Magnetic Structure Determination from Neutron Powder Diffraction Data

Transcription

1 Magnetic Structure Determination from Neutron Powder Diffraction Data The Cosener s House, Abingdon, 2-4 December 2002 Organiser: Paolo G. Radaelli (RAL-CLRC & UCL) Tutors: Andrew S. Wills (UCL) & Juan Rodriguez-Carvajal (LLB) Page / 42

2 Programme Thursday 2 December :00 (PGR) Welcome & Introduction 4:20 (PGR) Magnetic SG Symmetry Shubnikov Groups. 5:0 Coffee Break 5:40 (PGR) Generating magnetic structures from SG Defining a SG from a known magnetic structure. Additional topics in Shubnikov groups. 6:0 Break 6:20 (JRC) What is a Magnetic Structure? Description of Magnetic Structures Propagation Vectors 7:0 Break 7:20 (JRC) Examples of Common Magnetic Structures Additional topics 8:0 Close 9:00 Dinner 20:30 (All) Problem-Solving Session with Tutors: Analysis and Description of Magnetic Structures 22:00 Close Page 2 / 42

3 Programme (cont d) Friday 3 December :30 Breakfast 8:30 (ASW) Symmetry of Magnetic Structures Representation Analysis 09:30 Break 09:40 (ASW) Representation Analysis (con't) 0:30 Coffee Break :00 (ASW ) Introduction to Software for Symmetry Analysis: SARAH, etc. :50 (All) Additional topics/discussion 2:5 Lunch 3:30 (All) Training Session on Software for Symmetry Analysis 5:30 Coffee Break 5:50 (JRC) Unpolarised Neutron Scattering - Powder Diffraction 6:40 Break 6:50 (PGR) Software for magnetic PD: GSAS 7:40 (JRC) Software for magnetic PD: FULLPROF 8:30 Close 9:00 Christmas Dinner 20:30 (All) Training Session with Tutors: Refinements of Magnetic Structures 22:00 Close Page 3 / 42

4 Programme (cont d) Saturday 4 December :30 Breakfast 08:30 (Tutors) Additional software topics (if required) 09:00 (All) Training Session with Tutors: Refinements of Magnetic Structures (cont'd) 0:00 Coffee Break 0:30 PGR Powder Diffraction Instrumentation :0 Break :20 (JRC) Additional Topics: Polarised Neutrons, etc. 2:5 Lunch & Close Page 4 / 42

5 Page 5 / 42 Magnetic Symmetry - Shubnikov Groups Paolo G. Radaelli

6 Objectives of this module To learn the relevance of time reversal for magnetic structures. To learn how PG and SG operators act on spins. To learn how magnetic groups can be constructed from subgroups of index 2. To learn how to find those on the International Tables for PG and SG. To learn about magnetic lattices. To be able to construct invariant spin arrangements for magnetic SG, with specific examples. To learn the relation between Shubnikov groups and representations. Reference: W. Opechowski and R. Guccione, Magnetic Symmetry, in Magnetism, Vol II part A, ed. By G.T. Rado and H. Suhl. Academic Press (New York and London), 965, pp Page 6 / 42 Notation- Element of Space group {F}: F=(R τ(r)+t), where R is a proper or improper rotation, t is a primitive translation and τ(r) is a non-primitive translation. {R} is the point group associated with {F}. If {(R 0)} is a subgroup of {F}, then {F} is called symmorphic. Given a position r on the lattice, the subgroup {F (r)} for which (R τ(r)+t) r = t + r is called site space group, and its point group {R(r)}. We shall call {A}={E, E } the 2-elements group of the time identity (E) and time inversion (E ). Because crystal structures are static, {F} {A} is also a symmetry group of the crystal.

7 Notation-2 However, if we add spins (i.e., magnetic moments) to some of the atoms, time reversal will switch the direction of the spins. So {F} {A} cannot be a symmetry group of the magnetic structure, and the magnetic symmetry group, {M}, must be a subgroup of {F} {A}. In particular, (I E ) cannot belong to it. Forward time Backward time Purpose of the study of magnetic symmetry is to generate systematically all the magnetic groups associated with a particular space group of the crystal structure. Caveat Magnetic space groups, also known as Shubnikov groups, are perhaps the most elegant description of magnetic structures. However, in the presence of magnetic ordering, the crystallographic space group is often not known a priori, because the symmetry subtly is lowered by magnetic ordering itself. One has therefore to lower the symmetry in a systematic way, which is the purpose of representation theory. The study of Shubnikov groups with therefore serve as an introduction to the more general methods to be described in the remainder of the workshop. Page 7 / 42

8 Coloured groups We have just seen that the magnetic space group {M} must be a subgroup of {F} {A}, and cannot contain (I E ). However, it can contain elements of the form (F E ), which will be called primed (F ). If it does not, it is called a trivial (or colourless) group. Trivial groups can describe magnetic structures. Groups of the type {F} {A}are called gray or paramagnetic groups. All non-trivial subgroups of {F} {A} are called black and white groups. The original concepts and terminology were developed by Heesch (930) and later by Belov and by Zamorzaev (~955, including a complete list of the magnetic SG). The original aim was purely mathematical or crystallographic (study of coloured patterns on lattices, with A being colour inversion). The application to magnetism is due to Landau & Lifshitz (958). These concept can be extended to multicoloured SG, which are also of some interest for magnetism. Aleksei Vasil'evich Shubnikov was the founder and first director of IC-RAS. Colour vs. Spin The analogy between colour and spin can be made by replacing the meaning of E from time reversal to colour change. However, colour and spin differ fundamentally in the way the regular space group operators act upon them. Colours are scalars, whereas spins are axial vectors. It is important to remember that an axial vector is left invariant by centering. Therefore, proper rotations act on spins in the same way as on normal (polar) vectors, whereas for mirror operations and centering there is an additional spin flip. On top of this, priming any operator will entail and additional spin flip. Page 8 / 42

9 m x 2 z, 3 z, 4 z, 6 z Unprimed Flip s y s z Rotate s x,s y No effect No effect Primed Flip s x Rotate s x,s y Flip s x,s y,s z Flip s x,s y,s z Does not occur Constructive theorem We will give here the fundamental lemma to construct magnetic groups. It will apply equally well to SG, PG or lattices. Let {G} be a crystallographic group, {M} a derived magnetic group (subgroup of {G} {A}) and {G M } the group of the elements of {G} that are unprimed in {M}. It can be easily shown that {G}= {G M } +p {G M } where p does not belong to {G M }, which is therefore a subgroup of index 2 in {G}. This simply has to do with the fact that the product of 2 primed elements must be unprimed. Follows {M}= {(G M E)} +p {(G M E )} So, the problem of finding all magnetic groups arising from a crystallographic group {G} is reduced to that of finding all subgroups of index 2 of {G}. Page 9 / 42

10 Example: magnetic point groups To apply this rule to magnetic point groups, one needs to look no further that page 78 of the International Tables (copied overleaf). Subgroups of index 2 are those that have exactly half the number of elements of the original group. Elements of the subgroup will be unprimed, all the remaining elements being primed. Example : mmm 2 m 2 m 2 m m m 2 2 m m m m m m m m m m Page 0 / 42

11 Example 2: 4/m m m 4 m 2 m 2 m m 4 m m 4 m 2 m 2 m m m m m m m ' 2 2 m m m m m m m m m Admissible magnetic point groups A point group is called admissible if all its operators leave at least one spin component invariant. Admissible MPG are marked with an asterisk in OG, Table I. As we shall see,admissible point groups (AMPG) have two very important applications. The site symmetry of a magnetic atom must be a AMPG. A Ferromagnetic MSG must have a AMPG as its MPG. The second is a necessary but not sufficient condition for the MSP to support FM. The other condition is that its lattice is a trivial magnetic lattice (see below). Page / 42

12 Examples of admissible PG * * (any direction) 42m 4 2 m 4 2m 42 m * Spin along z Page 2 / 42

13 Things to remark about the 42m example Spin must be parallel to the 4-fold axis (always true except for 2-fold axes). 4 must be black. In fact, for spins, 4 = 4 If a spin is in a plane, that plane must be red. If a spin is perpendicular to a 2-fold axis, that axis must be red. Note that the central 2-fold axis of 4 or 4 is always black. This is because the product of two primed 45-degree rotations is an unprimed 90-degree rotation. Page 3 / 42

14 Magnetic Bravais Lattices The constructive theorem we have used to generate the magnetic point groups, based on the identification of subgroups of index 2, can be applied to generate magnetic lattices {T M } from Bravais lattices {T}. In general, a group of lattice translations generated by a set of primitive vectors a, a 2, a 3 has exactly seven subgroups of index 2. However, they do not always generate independent MBL, as some of them can be equivalent by interchange of the axes. Also, we are only interested in MBL that belong to the same holoedry of the original BL. In fact, as we shall see in the remainder, MSG either share the same lattice with the original SG (trivial ML) or the same point group (and therefore, necessarily, the same holoedry). 2a, b, c a, 2b, c a, b, 2c a, b+c, 2c 2a, b, a+c 2a, a+b, c 2a, a+b, a+c Page 4 / 42

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17 Magnetic Space Groups Once again, the constructive theorem, based on the identification of subgroups of index 2, can be applied to generate magnetic lattices {F M } from space groups {F}. The method to generate all the MSG systematically is explained in OG. We will limit ourselves to use the International Tables volume A. In there, for each SG, there is a list of minimal nonisomorphic subgroups (Types I, IIa and IIb), and minimal isomorphic subgroups of lowest index (Type IIc). The index is indicated in brackets (e.g., [2]). Therefore, each subgroup listed as [2] will generate a nontrivial magnetic space group. There are 42 of them in total, 9 of which are non-trivial. All SG except F23 and P2 3 generate at least non-trivial MSG. Rules to construct Magnetic Space Groups. Identify the subgroups of type I. They share the same lattice (trivial MBL) but have different PG, so they correspond to all the subgroups of index [2] of the associated PG (with multiplicity). For these, it is sufficient to prime the generators that correspond to missing operators. 2. Identify all the other subgroups of index 2 (IIa, IIb and III, no distinction). Then Identify the MBL based on the supercell, and write its symbol. For the Belov symbol (right column in OG), one simply need to complete the H-M symbol with that of the subgroup. For the OG symbol, the modified operators with respect to the original symbol will be primed (e.g. m->n=m ) Page 7 / 42

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26 Rules to construct invariant spin arrangements Define a magnetic space group generated by the SG of the crystal structure. Identify the magnetic site, and define its magnetic point symmetry. A graphic representation of the MG is useful. Check that the site MPG is admissible for at least one spin component. Otherwise, the MSG does not support any magnetic structure on that site. Pick one admissible component, and apply in turn all the MSG operators on that component, propagating it to all equivalent sites. Rules to determine the MSG from a given structure Check that the magnetic structure Γ is Shubnikov-compatible. This is easily done by applying the operators of the crystal space group {F} upon Γ (including lattice doublings). Γ is Shubnikov-compatible if and only if, the structure is either invariant ( ) or reversed ( -) for each and every F in {F}. Prime all the operators in {F} for which Γ is reversed, and identify the new primitive translations. This completes the process. Page 26 / 42

27 Shubnikov groups and representations To make a link with the more powerful representation analysis, we can simply think of how a magnetic structure Γ, which is invariant under a particular magnetic groups {F M }, will transform under the parent space group {F}. It is apparent that Γ will be invariant ( ) under the operators which are unprimed in {F M }, whereas all the spins will be switched ( ) for the operators that are primed in {F M }. In other words, the set of numbers or is a representation of {F} onto the linear space generated by Γ. We can easily prove that the reverse is also true. We can conclude the Shubnikov groups are equivalent to -dimensional real representations of {F}, with the invariant Γ being their basis sets. In general, if we relax the requirement for invariance of the crystal structure, there is no reason to prefer these to all the (infinite) others, whence the need for extending the analysis to the full expansion in irreducible representations. Page 27 / 42

28 P422 (No 89) Special position: 4o [.2.] x, ½, 0 Page 28 / 42

29 P42 2 Special position: 4o [.2.] x, ½, 0 P P 422 Page 29 / 42

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31 Pnma (Pbnm) G x s x s x c + + C y s y s y a 4 4 Special position: 4b [ ] 0,0, ½ A z s z s z y=0 y= ½ Pn m a (Pb n m ) s x s x c s y - s y a 4 4 s z s z Special position: 4b [ ] 0,0, ½ y=0 y= ½ Page 3 / 42

32 Pn m a (Pb n m ) C x s x s x c + - G y s y s y a 4 4 Special position: 4b [ ] 0,0, ½ F z s z s z y=0 y= ½ Pn ma (Pb n m) A x s x s x c + + F y s y + + s y a 4 4 Special position: 4b [ ] 0,0, ½ G z s z s z y=0 y= ½ Page 32 / 42

33 Pnm a (Pb nm ) F x s x s x c + - A y s y + + s y a 4 4 Special position: 4b [ ] 0,0, ½ C z s z s z y=0 y= ½ Pnma (Pb nm) s x s x c s y - s y a 4 4 s z s z Special position: 4b [ ] 0,0, ½ y=0 y= ½ Page 33 / 42

34 Pnm a (Pbnm ) s x s x c s y - s y a 4 4 Special position: 4b [ ] 0,0, ½ s z s z y=0 y= ½ Pn ma (Pbn m) s x s x c s y - s y a 4 4 Special position: 4b [ ] 0,0, ½ s z s z y=0 y= ½ Page 34 / 42

35 Page 35 / 42 Magnetic Structure Determination Juan Rodriguez-Carvajal

36 Magnetic Structure Determination. What can we learn from a magnetic structure refined from neutron powder diffraction? (Tutorial, overview,... ) Juan Rodríguez-Carvajal Laboratoire Léon Brillouin (CEA-CNRS) DRECAM-CEA/Saclay, France & Service de Physique Statistique, Magnétisme et Supraconductivité DRFMC-CEA/Grenoble, France December 2002 Workshop on Magnetic Structures (Abingdon ) Content: What s and why magnetic structures Formalism to describe magnetic structures The Rietveld method Magnetic neutron scattering Magnetic structure determination: Indexing: SuperCell Symmetry Analysis: BasIreps Simulated Annealing: FullProf Examples SIMBO and ENERMAG: programs to analyze exchange interactions. December 2002 Workshop on Magnetic Structures (Abingdon ) Page 36 / 42

37 Ions with intrinsic magnetic moments Atoms/ions with unpaired electrons Intra-atomic electron correlation Hund s rule: maximum S core m = g J J (rare earths) Ni 2+ m = g S S (transition metals) December 2002 Workshop on Magnetic Structures (Abingdon ) E What is a magnetic structure? () Paramagnetic state: Snapshot of magnetic moment configuration J S S ij ij i j S i 0 J ij December 2002 Workshop on Magnetic Structures (Abingdon ) Page 37 / 42

38 What is a magnetic structure? (2) Ordered state: Anti-ferromagnetic Small fluctuations (spin waves) of the static configuration E J S S ij ij i j S i 0 J ij Magnetic structure: Quasi-static configuration of magnetic moments December 2002 Workshop on Magnetic Structures (Abingdon ) Magnetic structures for what? Fundamental properties of condensed matter. Exchange interactions related to the electronic structure. The first step for determining the exchange interactions by inelastic neutron scattering Permanent magnet industry. Chemical substitutions controlling single ion anisotropy, strength of effective interactions, canting angles, etc: NdFeB materials, SmCo 5, hexaferrites, spinel ferrites. Spin electronics, thin films and mutilayers December 2002 Workshop on Magnetic Structures (Abingdon ) Page 38 / 42

39 Formalism to describe magnetic structures December 2002 Workshop on Magnetic Structures (Abingdon ) (0,0) a 4 y Geometric relation between magnetic moments of crystallographically equivalent atoms 2 b 3 R( 3, ) 4 a 3b a yb 2 (,3) 4 December 2002 Workshop on Magnetic Structures (Abingdon ) Page 39 / 42

40 Magnetic structures Magnetic moment of each atom: Fourier series m lj k S k exp 2 ikr j l Necessary condition for real m lj kj kj S - S Position vector of atom j in the unit cell l R lj Rl rj la l2b l3c x j a y j b z j c December 2002 Workshop on Magnetic Structures (Abingdon ) Examples of Fourier coefficients for simple magnetic structures The simplest case: Single propagation vector k = (0,0,0) m Sk exp 2 ikr S December 2002 Workshop on Magnetic Structures (Abingdon ) Page 40 / 42 lj j l kj k The magnetic structure may be described within the crystallographic unit cell Magnetic symmetry: conventional crystallography plus time reversal operator: crystallographic magnetic groups

41 m Examples of Fourier coefficients for simple magnetic structures Single propagation vector k=/2 H 2 lj Sk j exp ikrl Sk j - k n( l) REAL Fourier coefficients magnetic moments The magnetic symmetry may also be described using crystallographic magnetic space groups December 2002 Workshop on Magnetic Structures (Abingdon ) a b a = a m b=b m b m a m k= (/2, /2) k= (0, 0) December 2002 Workshop on Magnetic Structures (Abingdon ) Page 4 / 42

42 Fourier coefficients of sinusoidal structures k interior of the Brillouin zone (pair k, -k) Real S k, or imaginary component in the same direction as the real one m S exp( 2i kr ) S exp( 2i kr ) lj kj l -kj l S 2 m u exp( 2 i ) kj j j kj m mu cos2 ( kr ) lj j j l kj December 2002 Workshop on Magnetic Structures (Abingdon ) Sinusoidal magnetic structure k= (k x, k y ) December 2002 Workshop on Magnetic Structures (Abingdon ) Page 42 / 42

43 Fourier coefficients of helical structures k interior of the Brillouin zone Real component of S k perpendicular to the imaginary component Sk j muju j imvjv j exp( 2 i kj ) 2 m m u cos2 ( kr ) m v sin2 ( kr ) lj uj j l kj vj j l kj December 2002 Workshop on Magnetic Structures (Abingdon ) The Rietveld Method December 2002 Workshop on Magnetic Structures (Abingdon ) Page 43 / 42

44 A powder diffraction pattern can be recorded in numerical form for a discrete set of scattering angles, times of flight or energies. We will refer to this scattering variable as : T. The experimental powder diffraction pattern is usually given as two or three arrays : T, y,,..., i i i i n The profile can be modelled using the calculated counts: y ci at the ith step by summing the contribution from neighbouring Bragg reflections plus the background. December 2002 Workshop on Magnetic Structures (Abingdon ) Powder diffraction profile: scattering variable T: 2, TOF, Energy Bragg position T h y i zero y i -y ci Position i : T i December 2002 Workshop on Magnetic Structures (Abingdon ) Page 44 / 42

45 The profile of powder diffraction profilepatterns The model to calculate a powder diffraction pattern is: y Ih( T Th) b ci i i h ( xdx ) Profile function characterized by its full width at half maximum (FWHM=H) and shape parameters (, m,...) ( x) g( x) f ( x) instrumental intrinsic profile December 2002 Workshop on Magnetic Structures (Abingdon ) The profile of powder diffraction patterns y Ih( T Th) b i c i i h I I h h s ( x, ) hi m Contains structural information: atom positions, magnetic moments, etc Contains micro-structural information: inst. resolution, defects, crystallite size,... b b Background: noise, diffuse scattering,... i i b December 2002 Workshop on Magnetic Structures (Abingdon ) Page 45 / 42

46 The Rietveld Method consist of refining a crystal (and/or magnetic) structure by minimising the weighted squared difference between the observed and the calculated pattern against the parameter vector: 2 n i w y y wi ( ) 2 i i ci December 2002 Workshop on Magnetic Structures (Abingdon ) 2 i : is the variance of the "observation" y i 2 i Magnetic neutron scattering December 2002 Workshop on Magnetic Structures (Abingdon ) Page 46 / 42

47 Diffraction pattern of incommensurate magnetic structures Portion of reciprocal space h = H+k Magnetic reflections Nuclear reflections Magnetic reflections: indexed by a set of propagation vectors {k} H is a reciprocal vector of the crystallographic structure k is one of the propagation vectors of the magnetic structure ( k is reduced to the Brillouin zone) December 2002 Workshop on Magnetic Structures (Abingdon ) Setting up of magnetic ordering in a Tb-Pd-Sn compound December 2002 Workshop on Magnetic Structures (Abingdon ) Page 47 / 42

48 Magnetic structure of DyMn 6 Ge 6 Conical structure with two propagation vectors k=(0,0,0) k= =(0,0,)=(0,0, 0.65) Nuclear contribution in blue December 2002 Workshop on Magnetic Structures (Abingdon ) Magnetic scattering of neutrons k I =2/ u I kf =2/ uf Q= k F - k I Dipolar interaction ( n, m): vector scattering amplitude Q m Q am Q re f Qm 2 2 Q December 2002 Workshop on Magnetic Structures (Abingdon ) Page 48 / 42

49 Magnetic scattering of neutrons Q m Q am Q re f Qm p f 2 Qm 2 Q p= cm Q=Q e m 3 exp( ) f Q m r iqr d r Only the perpendicular component of m to Q=2h contributes to scattering m December 2002 Workshop on Magnetic Structures (Abingdon ) Magnetic Bragg scattering Intensity (non-polarised neutrons) I N N * M M * h h h h h Magnetic interaction vector M h em(h) e M(h) e (e M(h)) h h Hk Scattering vector e h December 2002 Workshop on Magnetic Structures (Abingdon ) Page 49 / 42

50 December 2002 Workshop on Magnetic Structures (Abingdon ) The magnetic structure factor: s j j s js n j j j j S i exp T f O p k k r t k H S h h M 2 k k S S js n n n C js n j j s n s n n j j j j i exp js C T f O p k k h r S h h M 2 December 2002 Workshop on Magnetic Structures (Abingdon ) Magnetic structure determination Page 50 / 42

51 LaMnO 3 : 50K and 50 K December 2002 Workshop on Magnetic Structures (Abingdon ) LaMnO 3 : magnetic scattering Magnetic reflections Difference: 50K -50K Thermal expansion December 2002 Workshop on Magnetic Structures (Abingdon ) Page 5 / 42

52 Steps for Magnetic structure determination from Neutron Powder Diffraction Propagation vector(s) SuperCell Peak positions of magnetic reflections Cell parameters Symmetry Analysis BasIreps Propagation vector Space Group Atom positions Magnetic structure solution FullProf (Simulated Annealing) Integrated intensities Coefficients of the atomic components of basis functions December 2002 Workshop on Magnetic Structures (Abingdon ) The Program SuperCell (distributed within WinPLOTR) Program: SuperCell (J.Rodríguez-Carvajal, LLB-December-998) This program can be used to index superstructure reflections from a powder diffraction pattern. The first approach consist in searching the best "magnetic unit cell" compatible with a set of observed SUPERSTRUCTURE lines in the powder diffraction pattern. If the first approach fails to give a suitable solution, the superstructure may be incommensurate and a direct search for the propagation vector and one of its harmonics have to be used. December 2002 Workshop on Magnetic Structures (Abingdon ) Page 52 / 42

53 LaMnO 3 : Extraction of magnetic integrated intensities Main magnetic reflections Propagation vector k=(0,0,0) magnetic cell=crystal cell December 2002 Workshop on Magnetic Structures (Abingdon ) December 2002 Workshop on Magnetic Structures (Abingdon )

54 BasIreps provides the basis functions (normal modes) of the irreducible representations of the wave-vector group G k m ljs exp k S k 2 ikr js l S kjs n C n S k n Basis Functions (constant vectors): js S k n js December 2002 Workshop on Magnetic Structures (Abingdon ) December 2002 Workshop on Magnetic Structures (Abingdon )

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56 k=(0,0,0), =, n=,2,3 =, s=, j=,2,3,4 Format for FullProf The constant 2 may be substituted by S k n js December 2002 Workshop on Magnetic Structures (Abingdon ) Example of magnetic structure in terms of basis functions of the irreducible representations of the propagation vector group () S kjs n December 2002 Workshop on Magnetic Structures (Abingdon ) Page 56 / 42 C n S k n js For LaMnO 3 this sum is reduced to three terms for each representation to be tested. Example, for representation =, dropping the superfluous indices k, s, : S C S j n n n 23,, j S C S C S C S S C S C S C S

57 Example of magnetic structure in terms of basis functions of the irreducible representations of the propagation vector group (2) S C S j n n n 23,, j S C,, C,, C,, C,C,C S C,, C,, C,, C, C,C S C,, C,, C,, C,C, C S C,, C,, C,, C, C, C December 2002 Workshop on Magnetic Structures (Abingdon ) Refinement of the magnetic structure: mode (0,A y,0) December 2002 Workshop on Magnetic Structures (Abingdon ) Page 57 / 42

58 Magnetic structure determination in complex systems: Simulating Annealing December 2002 Workshop on Magnetic Structures (Abingdon ) Direct space methods: Look directly for coefficients of the expansion: k S C S js kjs or components of S k and phases, explaining the experimental data Minimize a reliability factor with respect to the configuration vector CCCCC, 2, 3, 4, 5,... C m December 2002 Workshop on Magnetic Structures (Abingdon ) Page 58 / 42 N n n n 2 2 h h, R c G G m obs r calc r r

59 Behavior of parameters in Simulated Annealing runs.00 LiMn 2 O Phases (mod Ph_Mn2a Ph_Mn2a2 Ph_Mn2a3 Ph_Mn2a /T December 2002 Workshop on Magnetic Structures (Abingdon ).20 Average step Corana algorithm LiMn 2 O <Step> 0.60 <Step> t December 2002 Workshop on Magnetic Structures (Abingdon ) Page 59 / 42

60 Interpretation of the magnetic structure as the ground state (first ordered state) of a classical spin system Classical magnetic energy: E = - ij J ij S i S j First ordered state corresponds to the lowest eigenvalue of the Fourier matrix of the exchange interactions: ij (k) = - m J ij (R m ).exp{-2i kr m } December 2002 Workshop on Magnetic Structures (Abingdon ) Crystal structure of R 2 BaNiO 5 (R: Pr, Nd, Tb, Dy ) Ba R NiO 6 December 2002 Workshop on Magnetic Structures (Abingdon ) Page 60 / 42

61 Neutron powder diffraction pattern of Dy 2 BaNiO 5 at.5k (/2 0 /2) (/2 3/2) Dy 2 BaNiO 5 Intensity (a.u.) (/2 0 3/2) (/2 /2) (/2 0 7/2) (/2 2 3/2) (3/2 0 3/2) (3/2 3/2) (/2 2 7/2) (3/2 2 3/2) Magnetic reflections 0 Dy 2 O (degrees) December 2002 Workshop on Magnetic Structures (Abingdon ) a) 0 Dy 2 BaNiO 5 40 b) a ) b c Temperature (K) Space group: Immm The magnetic structure of all members of the family R 2 BaNiO 5 has the propagation vector k=(/2,0,/2) Ni ions are in a single Bravais sublattice at (000)(2a) site. The rare earth site (4j) generates two sublattices : (/2, 0, z) and 2: (-/2, 0, -z) in a primitive unit cell December 2002 Workshop on Magnetic Structures (Abingdon ) Page 6 / 42

62 December 2002 Workshop on Magnetic Structures (Abingdon ) J 2 J 4 R 3+ J 6 c b a J 3 J J 5 Ni 2+ December 2002 Workshop on Magnetic Structures (Abingdon ) Page 62 / 42

63 R 2 BaNiO 5 : Ni and 2 R ions in the primitive cell Exchange Fourier matrix 3 x 3, k=(x, Y, Z) J (k) = 2J cos 2X J 2 (k) = J 2 (k) * = J 2 (+ exp{2ix}) + 2J 3 cos Y exp{i(x+z)} J 3 (k) = J 3 (k) * = J 2 (+exp{2ix}) exp{2iz}+2j 3 cosk exp{i(x+z)} J 22 (k) = J 33 (k) = 2J 4 cos2x J 23 (k) = J 32 (k) * = 4J 5 cosx cosy exp{iz} + J 6 exp{2iz} If we neglect R-R interactions (J 4 =J 5 =J 6 =0) the eigenvalues are: (k)=0, 2,3 (k)= 2J cos 2X ± ( J 2 cos 2 2X + 4 J 32 {+cos 2 2Y}) /2 The energy is independent of Z, so no 3D order is possible with isotropic exchange neglecting R-R interactions. December 2002 Workshop on Magnetic Structures (Abingdon ) Phase diagram for the topology of MFePO 5 CuFePO5 Fe3 O3 O4 Fe2 O4 CuFePO5 P O4 c b a P O3 Cu4 O2 O4 P O4 O2 P P O3 O2 O2 O3 O4 Fe O Cu2 Cu4 Fe Cu P O2 Fe b a c Cu P O3 O2 December 2002 Workshop on Magnetic Structures (Abingdon ) Page63 / 42

64 Similar magnetic structures k=(0,0,0), two sites M and Fe of four sublattices G M +G Fe = (+ -+ -; ) December 2002 Workshop on Magnetic Structures (Abingdon ) Sublattice magnetization of three members of the family MFePO 5 Continuous curves: Adjusted by self-consistent Brillouin functions December 2002 Workshop on Magnetic Structures (Abingdon ) Page 64 / 42

65 The program SIMBO Geometrical analysis of the exchange paths for ionic structures (input: cell parameters, space group, atom positions in the asymmetric unit, magnetic moments) Automatic generation of the formal expression of the Fourier matrix of the exchange integrals Output file for ENERMAG 2 iy ( ) 2 iz ( ) 2 i( XZ) 2 i( XY) 2 iy ( ) 0 0 J2( e ) 0 Je Je 3 Je Je 3 2 iy ( ) J2( e ) J3 J J3 J 2 iy ( ) 2 i( Z) 2 i( XZ) 2 i( X) J2( e ) Je J3e Je J 3 2 iy ( ) 2 iy ( ) 2 iy ( ) 0 J2( e ) 0 0 J3 J J3e Je ( k) 2 i( Z) 2 i( Z) 2 i( X) Je J3 Je J3 0 J4( e ) i( XZ) 2 i( XZ) 2 i( X) Je 3 J Je 3 J J4( e ) i( XY) 2 i( X) 2 i( Y) 2 i( X) Je J3 Je Je J4( e ) 2 iy ( ) 2 iy ( ) 2 i( X) Je 3 J J3 Je 0 0 J4( e ) 0 December 2002 Workshop on Magnetic Structures (Abingdon ) The program ENERMAG ij (k) = - m J ij (R m ).exp{-2i kr m } The program handles the diagonalization of the Fourier matrix solving the parametric equation: (k, J) v(k, J)= (k, J) v(k, J) For a given set J ={J ij }, and no degeneracy, the lowest eigenvalue min (k 0, J) occurs for a particular k 0. The corresponding eigenvector v min (k 0, J) (that may be complex for incommensurate structures), describes the spin configuration of the first ordered state December 2002 Workshop on Magnetic Structures (Abingdon ) Page 65/ 42

66 The compounds of formula MFePO 5 can be modelled with four exchange interactions: 20 J corresponds to the exchange between M 2+ and Fe 3+ nearest neighbours. J 2 =-6.7 J 2 corresponds to the exchange between two M 2+ cations (double oxygen bridge). J G M +G Fe F M +F Fe J 3 corresponds to the exchange between next nearest neighbours M 2+ and Fe 3+ cations (single oxygen bridge). F M -F Fe G M -G Fe J 4 exchange between nearest neighbours Fe 3+ cations (single oxygen bridge), taken here as J 4 = J 3 20 December 2002 Workshop on Magnetic Structures (Abingdon ) 20 J 2 = -5 G M +G Fe F M +F Fe J Incommensurate F M -F Fe G M -G Fe December 2002 Workshop on Magnetic Structures (Abingdon ) Page 66 / 42 J 3 20

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71 Conclusions Magnetic Powder diffraction is the primary technique for determining magnetic structures. Sometimes diffraction alone is not sufficient to determine without ambiguities a unique solution, even using single crystals. The Rietveld method extended for incommensurate structures has powered the powder method, providing not only the average structure but also the correlation lengths along different crystallographic directions. The analysis of exchange paths and the use of classical models for studying the magnetic ordering gives a first estimation of the relative exchange interactions. December 2002 Workshop on Magnetic Structures (Abingdon ) Page 7 / 42

72 THE PHASE PROBLEM OF MAGNETIC STRUCTURES WITH NON SYMMETRY-RELATED PROPAGATION VECTORS Juan Rodrguez-Carvajal Laboratoire Leon Brillouin, C.E.A./C.N.R.S., F-99-Gif-sur-Yvette CEDEX, France. Introduction ABSTRACT In these notes the limitations of neutron diraction for determining the true magnetic structure of some compounds are discussed. The analytical expressions linking the measurable quantities to the model of a magnetic structure do not contain a crucial parameter: the phase factor between two Fourier coecients not related by symmetry. The impossibility to obtain this parameter by conventional methods precludes the access to the true spin arrangement in the solid. The problem is rst formulated analytically and illustrated by some simple examples, secondly we shall present some real examples concerning incommensurate-to-commensurate magnetic phase transitions and, nally, some conclusions are stated It is frequent the discovery of magnetic compounds that exhibit more than one propagation vector. The typical case is the so called multi-k structures, observed in some intermetallic compounds of high crystallographic symmetry. Multi-k structures refers to a magnetic structure in which more than one arm of the star of k participates into the actual spin arrangement. That is, the transition chanel, in terms of the Izyumov's school 2, has more than one propagation vector. Symmetry relations between the Fourier coecients of the magnetic structure, when all the propagation vectors belong to a single star, can be obtained by group theory using the geometrical method of Bertaut 3 or the algebraic straightforward expressions that have been given by Izyumov and collaborators 2. The practical determination of the transition chanel could be dicult because the magnetic phase transition, and the concommitant domain formation, produces satellites (in a single crystal diraction pattern) which are not distinguishable (in usual conditions) from those of a true multi-k structure. External elds have to be applied to decide what is the actual situation. More unusual is the case showing two propagation vectors not belonging to the same star. However, a well known case is particularly common: the conical structures. Nagamiya 4 has given the conditions for two independent propagation vectors to describe constant moment (CM) magnetic structures. Nagamiya treated combinations of propagation vectors of the form k ==2H (or k ==4H, k =3q) and k 2 = q at the interior of the Brillouin Zone (k 2 2 IBZ), so that the relative orientation of the Fourier coecients is xed and the relative phase is irrelevant. In this paper we shall formulate the problem in its full generality in relation with the practical structure determination. For that a summary of the most important scattering formulas is rst given. 2. Neutron Scattering Cross Sections and Magnetic Structure Factor For polarized neutrons the total scattered intensity and the nal polarisation of scattered neutrons for the scattering vector h is given by the Blume's equations 5. The scattered intensity is: I h = N h N h + N hfp M?h g + N h fp M?hg + M?h M?h + ip fm?h M?h g () In this paper we use the terms spin and magnetic moment indistinctly. The term spin arrangement is also used as synonymous of magnetic structure Page 72 / 42

73 The equation dening the scattered polarisation is: P s I h = PN h N h + N h M?h + N hm?h + ip fm?hn h M?hN h g + M?hfP M?h g + M?h fp M?hg PfM?h M?h g + ifm?h M?h g (2) Where P and P s are the incident and scattered neutron polarisation, N h is the nuclear structure factor and M?h is the magnetic interaction vector dened as: M?h = e (M(h) e) =M(h) (e M(h))e (3) M(h) is the magnetic structure factor, and e is the unit vector along the scattering vector h. The scattering vector is h=h+k where H is a reciprocal lattice vector of the crystal structure and k the propagation vector corresponding to the current magnetic reection. For a pure magnetic reection N h =0 The magnetic structures that we are considering have a distribution of magnetic moments that can be expanded as a Fourier series: m lj = X fkg S kj expf2ikr l g (4) The sum is extended to all propagation vectors that could belong to dierent stars. The Fourier coecients S kj are, in general, complex vectors. The magnetic structure factor can be written as: M(H + k) =p Xn c j= f j (H + k)s kj expf2i(h + k)r j g (5) The sum is over all the magnetic atoms in the crystallographic cell. The constant p(= r e =2) is and allows the conversion of the Fourier components of magnetic moments, given in Bohr magnetons ( B ) to scattering lengths units of 0 2 cm. f j (H+k) is the magnetic form factor and r j is the vector position of atom j. In the above expression the atoms have been considered at rest. If thermal motion is considered and if symmetry relations are established for coupling the dierent Fourier components, we obtain the general expression of the magnetic structure factor: M(h) =p nx j= O j f j (h)t j (iso) X s M js S kj T js expf2i[(h + k)fs j tg s r j kjs ]g (6) The sum over j concerns the atoms of the magnetic asymmetric unit for the wavevector k (the Fourier component with index k contributes only to the k-satellite). So that j labels dierent sites. The anisotropic temperature factor, T js, is not generally necessary to be used in magnetic renements(t js = ). The sum over s concerns the dierent symmetry operators of the crystal space group that belong to the wave vector group. The matrix M js transform the components of the Fourier term S kj = S kj of the starting atom j to that numbered as js in the orbit of j. The phase factor kjs has two components: kjs = kj + kjs (7) kj is a phase factor which is not determined by symmetry. It is a renable parameter and it is signicant only for an independent set of magnetic atoms (one orbit) which respect to another one. kjs is a phase factor determined by symmetry. The Fourier component k of the magnetic moment ofatomj, S kj, is transformed to S kjs = M js S kj expf2i kjs g (8) 2 Page 73 / 42

74 The matrices M js and phases kjs can be deduced from the atomic basis functions, obtained by applying projection operator formulas, corresponding to the active representation(s) participating in the denition of the actual magnetic structure. The sign of kjs changes for -k. In the general case S kj is a complex vector with six components. These six components per magnetic orbit constitute the parameters that have to be rened from the diraction data. Symmetry reduces the number of free parameters per orbit to be rened. An alternative expression of the magnetic structure factor can be written as a function of mixing coecients (parameters to be rened) and the atomic components of the basis functions of the relevant representation(s) 2. In the case of a commensurate magnetic structure one can calculate the magnetic structure factor in the magnetic unit cell. In such a case S kj are real vectors corresponding to the magnetic moment of the atom j, the matrices M js are real and all phases verify kjs = 0. The crystallographic magnetic groups theory can be fully applied in such a case 6. If the magnetic structure represents an helical order the Fourier coecients are of the form: S kj = 2 [m ju j + im2jv j ]expf2i kj g (9) where u j and v j are orthogonal unit vectors. If mj = m2j = m0 the magnetic structure for the sublattice j corresponds to a classical helix (or spiral) of cylindrical envelope. All j atoms have a magnetic moment equal to m0. If mj 6= m2j the helix has an elliptical envelope and the moments have values between min(mjm2j) andmax(mjm2j). If = 0 the magnetic structure corresponds to a modulated sinusoid of amplitude mj. m2j 3. The phase between independent k-vectors When more than two independent propagation vectors appears in the diraction pattern, the analysis of the data is unable to give a unique answer to the problem of the magnetic structure. In general is not possible to discriminate between the presence of two magnetic phases co-existing in the crystal and a coherent superposition of these two magnetic structures. We shall be concerned only with the latter picture. Even from this hypothesis it is not possible to get uniqueness. This can be seen adding a phase factor, depending only on k, to the Fourier series equation (4): m lj = X fkg S kj expf2i(kr l + k)g (0) The magnetic structure factor [equation (5)] transforms to: Xnc M(H + k) =p expf2i kg f j (H + k)s kj expf2i(h + k)r j g () j= The phase k appears in the expresion of the magnetic structure factor as a multiplicative phase factor that does not change the intensity of equation () or the scattered polarisation of (2) for a pure magnetic reection. The phases k are not accesible experimentally, so the real magnetic structure cannot be obtained from diraction measurements alone. The most simple case in which the phase plays an important role is the sinusoidally modulated structure in a simple Bravais lattice (a single magnetic atom per primitive cell) when the propagation vector takes special values. The Fourier coecient and the corresponding magnetic moment at cell l are: S k = 2 m ou expf2i kg m l = m o u cos 2(kR l + k) 3 Page 74 / 42

75 The phase k plays no role when k 2 IBZ and has no rational components. A change in the phase has the same eect as a change of the origin in the whole crystal. All magnetic moments between m o u and m o u are realized somewhere in the lattice. However, if k = =4H and k = =8 the magnetic structure is a CM-structure with the sequence f++++:::g. This structure is indistinguishable of the sinusoidally modulated structure obtained with an arbitrary valueof k. If all the components of k are rational the selection of the phase can have important consequences for the spin arrangement. This is the simplest case in which the physical picture depends on the election of a parameter ( k) that is not accessible by diraction methods. Physical considerations lead us to prefer one model among several other. For instance, CM-structures are normally expected at vey low temperatures when magnetic atoms have anintrinsic magnetic moment. This condition reduces the number of ways to combine non symmetry-related propagation vectors to several specic cases that have been discussed by Nagamiya 4. Let us discuss some unusual simple cases that will be illustrated with real examples. 4. Fluctuating magnetic structures The magnetic structures with more than one pair (k,-k) of propagation vectors not satisfying the Nagamiya's conditions are, as is the sinusoidally modulated magnetic structure, general non-constant moment structures. We shall call these spin congurations: uctuating structures ỵ Fluctuating Structures with irrelevant phase-factors This case corresponds to the combination of k = =2H and q 2 IBZ vectors. To simplify the notation we shall treat only one of the atoms of a particular Wycko site and we drop the reference index. The propagation vector q describes a helical conguration, and k corresponds to a uniaxial antiferromagnetic conguration, so that the Fourier coecients of the atom are: S q = 2 m [u + iv] expf2i qg S k = m 2 n where, as above, u and v are orthogonal unit vectors dening the plane of the spiral of axis w = u v, andn isaunitvector dening the axis of the spin conguration related to propagation vector k ==2H. The director cosines of n with respect to the axes ( u v w) are(n n 2 n 3 ). The magnetic moment distribution of a coherent superposition of the two types of Fourier coecients is given by the following formula (notice that l =2(qR l + q) andl h = HR l ): m l = m cos 2(qR l + q)u + m sin 2(qR l + q)v + m 2 expfihr l gn = m cos l u + m sin l v + m 2 () l h n = (m cos l +() lh m 2 n )u +(m sin l +() lh m 2 n 2 )v +() lh m 2 n 3 w(2) The modulus of the magnetic moment can be calculated by taking the square of equation (2): m 2 l = m 2 + m m m 2 () l h (n cos l + n 2 sin l ) = m 2 + m m m 2 () l h cos l (3) If n is parallel to w the moment is constant and we obtain an antiferromagnetic conical structure (if H= 0, we obtain the classical ferromagnetic conical structure). For the y The term uctuating has no dynamic content in the present context 4 Page 75 / 42

76 general orientation of n (non vanishing components in the u-v plane) the modulus of this q distribution is not constant. q The amplitude varies between the two extreme values m 2 + m m m 2 sin and m 2 + m 2 2 2m m 2 sin, being the angle of n with w. A real system in which this behaviour seems to take place is the compound CsMnF Another interesting system is TbMn 6 Ge ẓ 6 The second wavevector, in this case, is k =0 and the associated magnetic moment lies within the u-v plane dening the spiral plane of the rst propagation vector. This gives rise to a distorted spiral structure. In all these cases, the selection of the phase factor q is completely irrelevant. That is, the physical picture obtained after using the equation (0) is not changed by varying the phase factor. Fluctuating Structures Approaching CM-structures We shall now consider the case of two pairs of propagation vectors (k,-k) and (q,-q) verifying k q 2 IBZ.Such a magnetic structure has as Fourier coecients: S k = 2 (R k + ii k ) S q = 2 (R q + ii q )expfi g Using the notation kl =2kR l the magnetic moment distribution is given by: m l = R k cos kl + I k sin kl + R q cos( ql + )+I q sin( ql + ) (4) This moment distribution is generally a non-cm structure and the change of the phase factor can modify the physical picture if both vectors k and q have rational components. This last case is interesting when the components are simple integer fractions because one can treat the problem using the magnetic cell and search for a magnetic space group that x automatically the phase. The nding of such a commensurate magnetic structure does not eliminate the problem of uniqueness of the magnetic moment distribution compatible with the experimental results. However, the possibility tohave a simple spin arrangement with magnetic moments of atoms approaching the expected intrinsic moment is more satisfying form the physical point of view. If a CM-structure can be found rening the magnetic structure using the magnetic cell, a particular set of equations (4) can be established for atoms inside the magnetic cell and the phase factor can be obtained solving these equations. Of course, to get a set of compatible equations the vectors R and I cannot be arbitrary. An example can be readily shown if we consider only real Fourier coecients in equation (4). We can write for the -component: R q cos( ql + )=m l R k cos kl =) =cos m l R k cos kl R q ql The above equations must be veried for the set of points l inside the magnetic cell and for all components simultaneously. This indicates that only very special relationships between Fourier coecients must be veried to have a single to connect the two descriptions. An interesting example is the magnetic ordering of TbGe 9 3. This compound crystallizes in the space group Cmcm,(a =4:07b =20:8c =3:92 A), with Tb-atoms in positions (4c) (0y=4). Below the Neel temperature (T N = 40K) the magnetic order is characterized by two independent propagations vectors k =(k x 0 0) and q =(q x 0q z ) with k x q x and 2 q z. Below 3 T ic = 24K the propagation vectors lock-in to commensurate values. Both vectors verify k q 2 IBZ with a two-arm star for k z See also the article: Magnetic Spiral Structures in the Hexagonal RMn 6 Ge 6 Compounds, by P. Schobinger-Papamantellos, J. Rodrguez-Carvajal, G. Andre and K.H.J. Buschow, in these proceedings 5 Page 76 / 42

77 (G k = C2cm) and a four-arm star for q (G q = Cc). The renement of the magnetic structure at low temperature in the magnetic unit cell using powder diraction data provides a quasi-collinear structure with two types of Tb-atoms having similar moments (m(tb )=9:2 B m(tb 2 )=8:8 B ). The renement using real Fourier coecients for all propagation vectors (including the second pair of the star of q) gave similar agreement. A systematic search of the phase factors using a computer program 0 allows the nding of a consistent set of phases that produces uctuations of m(tb)between 9:4 B and 7:0 B. The spin arrangement is similar to that observed in the magnetic cell renement. For the incommensurate phase we suppose that the spin arrangement does not change dramatically, so that the phases found for the lock-in phase are still valid. Symmetry analysis can be applied to each wave vector separately. There is no interference terms between reections belonging to dierent sets of satellites, so that we can proceed as if two magnetic dierent phases co-exist and only at the end of the analysis we can think in the coherent superposition of both phases. The computer program 0 we have written can be used as a general tool for searching phase factors between Fourier coecients belonging to non-symmetry related wave vectors giving the lowest uctuation between m min and m max. 5. Conclusions The physical origin of the stabilization of two propagation vectors belonging to dierent stars is not yet clear in the absence of external elds. In Bravais lattices we have to think in the action of higher order terms (biquadratic) in the spin hamiltonian to stabilize two propagation vectors. In complex crystal structures the nature of the ground state is not known in the general case and, probably, it is not necessary to invoke higher order terms to stabilize two non-related propagation vectors. Only the case of conical structures (k = 0 and q 2 IBZ) has been studied with some detail for the spinel lattice. We can conclude that only a physical model based in the microscopic spin-spin interactions is able to x completely the phases appearing in the Fourier expansion of the magnetic moment distribution in the solid. Experimentally, other techniques (like Mossbauer spectroscopy, neutron or X-ray topography, -SR, etc...) may help, in some cases, to distinguish between several models. Unfortunately there is no general method to overcome this phase problem. References. J.Rossat-Mignod, Magnetic Structures, inmethods of Experimental Physics: Neutron Scattering, Vol 3, Academic Press (987). 2. Y.A. Izyumov, V.E. Naish, J.Magn.Magn.Mat. 2, 239 (979) Y.A. Izyumov, V.E. Naish and V.N. Syromiatnikov, J.Magn.Magn.Mat. 2, 249 (979) Y.A. Izyumov, V.E. Naish and S.B. Petrov, J.Magn.Magn.Mat. 3, 267(979) ibidem 3, 275 (979) 3. E.F. Bertaut, Spin Congurations of Ionic Structures. Theory and Practice, in Magnetism (G.T. Rado and H. Shul, eds.), Vol III, Ch. 4, 49 Academic Press, New York. (963) see also Acta Cryst A24, 27 (968), J. de Physique, Coll. C, sup. n-3,32, C-462 (97) and J.Magn.Magn.Mat. 24, 267 (98). 4. T. Nagamiya, Solid State Physics 20, (967) 5. M. Blume, Phys Rev 30(5), 670 (963) 6. W. Opechowski and R. Guccione, Magnetic Symmetry, inmagnetism (G.T. Rado and H. Shull, eds.), Vol II A, Ch. 3, 05 Academic Press, New York. (965). 7. J. Rodrguez-Carvajal, Physica B92, 55 (993) 8. P. Schobinger-Papamantellos, J. Rodrguez-Carvajal, G. Andre and K.H.J. Buschow, J.Magn.Magn.Mat. 50, 3 (995) 9. P. Schobinger-Papamantellos, J. Rodrguez-Carvajal, T. Janssen and K.H.J. Buschow, Proc.Int.Conf Aperiodic Crystals, p. 3 (994). Ed. G. Chapuis, W. Paciorek, World Scientic, Singapore. 0. J. Rodrguez-Carvajal, Program MOMENT (994), (unpublished).. D.H. Lyons, T.A. Kaplan, K. Dwight and.n. Menyuk Phys Rev26(2), 540 (962). 6 Page 77 / 42

78 MAGNETIC STRUCTURE DETERMINATION FROM POWDER DIFFRACTION USING THE PROGRAM FullProf JUAN RODRIGUEZ-CARVAJAL Laboratoire Léon Brillouin, (CEA-CNRS), CEA/Saclay, 99 Gif sur Yvette Cedex, FRANCE In this paper the techniques for magnetic structure determination from neutron powder diffraction (NPD) data as implemented in the program FullProf are reviewed. In the general case the magnetic moment of an atom in the crystal is given as a Fourier series. The Fourier coefficients are complex vectors constituting the unknowns to be determined. These vectors define the magnetic structure and they correspond to the atom positions of an unknown crystal structure. The use of group theoretical methods for the symmetry analysis is needed to find the smallest set of free parameters. In general the Fourier coefficients are linear combinations of the basis functions of the irreducible representations of the wave vector group. The coefficients of the linear combinations can be determined by the simulated annealing (SA) technique comparing the calculated versus the observed magnetic intensities. The SA method has been improved and extended to the case of incommensurate magnetic structures within FullProf. Introduction In the last years the Rietveld Method (RM) has allowed great progress in the analysis of powder diffraction data. The RM is not designed for structure determination, it is just a least squares optimisation of an initial model of the crystal and magnetic structure supposed to describe approximately the experimental powder diffraction pattern. It is important to start with a good initial model in order to succeed the refinement procedure. In this paper we shall be concerned with the problem of getting the initial model of a magnetic structure in order to refine it from powder diffraction data. We shall describe the basis of the technique and the way the magnetic structure determination is implemented in the program FullProf. 2 The formalism of propagation vectors for describing magnetic structures. The reader interested in the basis of the elastic magnetic scattering in relation with magnetic structures may consult the references [, 2]. Here we will follow the reference [3] but using a different convention for the sign of phases and a somewhat different notation. The intensity of a Bragg reflection (we neglect here the geometrical factors) for non polarised neutrons is given by: * * I h N h N h M h M h () where N h is the nuclear structure factor and the magnetic interaction vector M h is defined as: M h e Mh e Mh ee Mh (2) M(h) is the magnetic structure factor, and e is the unit vector along the scattering vector h=h+k, where H is a reciprocal lattice vector of the crystal structure and k the propagation vector corresponding to the current magnetic reflection. The magnetic structures that we are considering have a distribution of magnetic moments that can be expanded as a Fourier series: Page 78 / 42

79 mlj S k j exp 2 ikrl (3) k The sum is extended to all propagation vectors that could belong to different stars. The Fourier coefficients S kj are, in general, complex vectors. The magnetic structure factor corresponding to such a magnetic structure can be written as: n iso poj f j Tj M js kjexp 2 i S j (4) kjs M h h S H k t r s j s The sum over j concerns the atoms of the magnetic asymmetric unit for the wave vector k. We are concerned only with magnetic atoms within the crystallographic unit cell, so that j label different sites: f j (h) is the magnetic form factor and r j is the vector position of atom j. The constant p = r e /2 = allows the conversion of the Fourier components of magnetic moments, given in Bohr magnetons to scattering lengths units of 0-2 cm. The sum over s concerns the different symmetry operators of the crystal space group that belong to the wave vector group G k (subgroup of the crystallographic space group formed by the operators leaving invariant the propagation vector). The matrix M js transform the components of the Fourier term S kj of the starting atom j to that numbered as js in the orbit of j. The phase factor kjs has two components: kjs k j k js (5) kj is a phase factor that is not determined by symmetry. It is a free parameter and it is significant only for an independent set of magnetic atoms (one orbit) which respect to another one. kjs is a phase factor determined by symmetry. The Fourier component S kj of the representative starting atom j is transformed to Sk js M jss k j exp 2 ik js (6) The matrices M js and phases kjs can be deduced from the atomic basis functions, obtained by applying projection operator formulas, corresponding to the active representation(s) participating in the definition of the actual magnetic structure. The sign of kjs changes for -k. In the general case S kj is a complex vector with six components. These six components per magnetic orbit constitute the parameters that have to be refined from the diffraction data. Symmetry reduces the number of free parameters to be refined. In some cases, transformations like expression (6) cannot be obtained from the basis functions of the irreducible representations of the propagation vector group; for those cases an alternative expression of the magnetic structure factor can be written as a function of "mixing coefficients" (parameters to be refined) and the atomic components of the basis functions of the relevant representation [4]. The expression of the Fourier coefficients in terms of the atomic components of the basis functions is given as: k Sk js C jm S m js (7) m The formula of the magnetic structure factor is then transformed to: n iso k p O jf j Tj C jm m jsexp2i s j k j M h h S h r j m s (8) In the above expressions, labels the active irreducible representation,, of the of the propagation vector group G k, labels the component corresponding to the dimension of the representation, m is an index running between one and the number of times the k S js representation is contained in the global magnetic representation M. Finally n Page 79 / 42

80 are constant vectors obtained by the application of the projection operator formula to unit vectors along the directions of the unit cell basis. An addition sum over is sometimes necessary when more than one irreducible representation is involved in the magnetic phase transition. See reference [4] for details. If the magnetic structure has several propagation vectors k, it is not possible in general to determine unambiguously the spin configuration, because the phase between the different Fourier components cannot be determined. Fortunately, nature often selects simple solutions and many magnetic structures have a single propagation vector, or display some symmetry constraints that reduce the complexity of the periodic magnetic structure given by Eq.3. Solving a magnetic structure consist of finding a set of propagation vectors indexing the whole set of magnetic reflections and a set of mixing coefficients (or, equivalently, the components of the Fourier coefficients and phases) providing a good agreement between the intensities of the observed and calculated (using the above expressions) magnetic reflections. In some cases the search for a good starting model may be formulated in terms of other set of parameters. For instance, in cases of conical/helical structures involving magnetic atoms with a common cone-axis, the magnetic structure factor can be written in terms of the module of the magnetic moments, the angle between the moments and the cone-axis, and phases between the different atoms. This description in real space gives a more intuitive picture of the magnetic structure. 3 The search for the propagation vector and symmetry analysis. The first problem to be solved before attempting the resolution of the magnetic structure is the determination of the propagation vector(s), i.e. its periodicity. To find k is necessary to index the magnetic reflections appearing below the ordering temperature. With a single crystal the task is somewhat easy, but is tedious for a powder because only the module of reciprocal vectors is available. We have developed a method for searching the propagation vector of a commensurate or incommensurate structure implemented in the program SuperCell [5]. Once an approximate propagation vector is obtained the symmetry analysis according to references [4] can be started. The program BasIreps may k be used for obtaining the vectors S n js in Eq.7 for each crystallographic site occupied by magnetic atoms. To solve the magnetic structure, the integrated intensities of the magnetic reflections may be obtained using the method of profile matching, simultaneously with the Rietveld method, implemented in the program FullProf [3, 5]. This mixed procedure has to be used with caution: no structural parameter of the known phase must be refined. This is the usual case of neutron diffraction patterns of magnetically ordered compounds, where the nuclear reflections coexist with the magnetic reflections. For illustration purposes we show in Fig. the plot of the observed versus calculated pattern of a portion of the simulated diffraction pattern of DyMn 6 Ge 6 at low temperature after performing the extraction procedure. The magnetic structure has two propagation vectors k =(0,0,0) and k 2 =(0,0,), with 0.65 with respect to the reciprocal lattice of the crystallographic unit cell. All satellite reflections are indexed with h=hk 2. There are also contributions to the same positions of the nuclear reflections, h=h (k =0), accounting for a ferromagnetic component. The spin arrangement corresponds to a double cone magnetic structure. Page 80 / 42

81 Intensity (arbitrary units) DyMn 6 Ge 6 Satellites Ferromagnetic contributions Figure. Profile matching refinement of the DyMn 6 Ge 6 neutron diffraction pattern at low temperature. The profile of the calculated nuclear contribution (upper reflection marks) is also displayed as a thick continuous line. The second set of reflection markers corresponds to the magnetic peaks. Markers at the same positions as the nuclear (first set) reflections correspond to k =(0,0,0), the extra markers are the position of the satellites corresponding to k 2 =(0,0,). 4 The resolution of magnetic structures from powder data: the simulated annealing method We shall describe the Simulated Annealing (SA) technique to solve the magnetic structure using clusters of overlapped reflections as single observations. The merging of clusters is automatically performed using the option profile matching of the program FullProf [5]. The SA method described below is also valid for the analysis of single crystal data where, except for domains, there is no reflection overlap. The SA algorithm is a general-purpose optimisation technique for large combinatorial problems introduced in 983 by Kirpatrick, Gelatt and Vecchi [6]. The function, E() to be optimised with respect to the configuration described by the vector state is called the cost function. In the context of magnetic structures the configuration is the list of all the components of the Fourier coefficients of magnetic atoms existing in the chemical unit cell and this list is obtained from the independent parameters ß that are those really participating in the annealing procedure. The most general case of parameters constituting the vector ß corresponds to the set of mixing coefficients of the linear combination given by Eq.8, but, as stated above, another set of parameters in real space (moment amplitudes, angles, ) may also be used. First we select an initial configuration, old, then each step of SA method consists of a slight change of the old configuration to a new one, new. If =E( new )-E( old ) 0 the new configuration serves as old configuration for the next step. If is positive, new is accepted as current configuration only with certain probability that depends on the so-called temperature, T, parameter and. The probability, given by the Boltzman factor exp(-), that a worse configuration is accepted is slowly decreased on cooling. Page 8 / 42

82 For magnetic structure determination, the cost function can be chosen as the conventional crystallographic R-factor, or some function related to it. In the new version of FullProf [5] the following expression is used: E[(ß)]= R[(ß)] = c k I obs (k)- S j(k) I calc (j)[(ß)] The sum over k is extended for all the observations (clusters of overlapped reflections), and that over j(k) for all the reflections contributing to the observation k. The constant factor c is given by: /c=i T = k I obs (k). S is a scale factor. To start solving a magnetic structure with the SA method one has to 00 create the intensity file where the indices of each 80 reflection and its intensity R-Factor in DyMn Ge 6 6 are written. This is 60 performed automatically within FullProf by using 40 profile matching modes and the option that outputs 20 the overlapped reflection clusters in a file that can be used as input for the SA Sequence Order method. The usual PCR Figure 2. Evolution of the cost function for accepted configuration in file [5] of FullProf is then the resolution of the magnetic structure of DyMn 6 Ge 6 by simulated used for controlling the annealing as a function of the sequential order of temperatures. For a single temperature on can see the dispersion of the R-factor, algorithm. A pseudo-code corresponding to the different configurations, that is reducing as describing the SA temperature decreases. procedure was given in reference [3]. The SA parameters are those defining the limits of loops in the algorithm described in [3]: T_ini = initial temperature, N = maximum number of temperatures, NcyclM = number of Montecarlo cycles per temperature, Accept=Minimum percentage of accepted configurations; and the cooling schedule T(t+)=qT(t) (q<, q 0.9). The user may select either a fixed step for each variable (that are defined within a simulation box of hard or periodic limits) or a variable step (Corana s algorithm) that is dynamically adapted in order to have an adequate rate of accepted configurations for each temperature [7]. The starting point may be an arbitrary configuration or a given one. At variance with least-squares optimisation methods, the SA algorithm never diverges. Always the algorithm proceeds roughly in two steps. The first step, at high temperatures, the algorithm is searching for the basin of attraction of the minimum in the configuration space, this part constitutes the magnetic structure determination. Once the region is attained, a more or less sharp drop in the average energy (R-factor) occurs. Then, the second step starts when the average R-factor is low enough, the algorithm enters in its phase of refinement, where the good configuration has already been found, and performs a progressive improvement of the solution. This is clearly seen in the behaviour of the cost function versus the ordinal number of the temperature parameter in Fig.2, illustrating the case of DyMn 6 Ge 6. In figure 3 it is shown the behavior of the amplitude of the magnetic moments of Dy and Mn atoms. The plot shows that there are a large Cost Function (R-Factor) Page 82 / 42

83 dispersion at the stage of magnetic structure solution (starting phase of the algorithm) and a progress toward definite values within the refinement region. For a given set of constraints the final average R factor should be reasonably good (below 20%) except for contradictory or false constraints. False minima are encountered when the number of free parameters is of the same order of magnitude than the number of 0 observations and/or the observations are of bad 8 quality (very weak magnetic reflections and 6 large errors associated to Amplitude of the magnetic them). Ambiguities can 4 moments of Dy and Mn atoms be easily discovered. When the intensity data 2 do not depend on a parameter, this shows an anomalous behaviour: in Sequence Order a plot similar to that of Fig. 3, large oscillations Figure 3. Evolution of the magnetic moment of Dy and Mn versus the number of sequential temperature. Similar plots can be observed for persist even at low other magnetic parameters (cone angles and magnetic phase angles). temperature. In conclusion, we have shown that the SA algorithm can be used for the magnetic structure determination even in the case of complex incommensurate magnetic structures. The method is straightforward and is fully implemented in the program FullProf that is publicly available [5]. Magnetic Moment ( B ) References. J.Rossat-Mignod, Magnetic Structures, in Methods of Experimental Physics: Neutron Scattering (987), Vol 3, Academic Press, New York. 2. P.J. Brown, Magnetic Structure, in Frontiers of Neutron Scattering, (986), p. 3, (R.J. Birgeneau, D.E. Moncton & A. Zeilinger eds.). Elsevier Science Publishers, North-Holland Amsterdam. 3. J. Rodríguez-Carvajal, Physica B 92 (993), p Y.A. Izyumov, V.E. Naish, J Magn.Magn.Mat. 2 (979), p. 239; Y.A. Izyumov, V.E. Naish and V.N. Syromiatnikov J Magn.Magn.Mat. 2 (979), p. 249; Y.A. Izyumov, V.E. Naish and S.B. Petrov, J Magn.Magn.Mat. 3 (979), p. 267; ibid. 3 (979), p. 275;Y.A. Izyumov, J Magn.Magn.Mat. 2 (980), p J. Rodríguez-Carvajal and T. Roisnel (999). FullProf, WinPLOTR and accompanying programs at 6. S. Kirpatrick, C.D. Gelatt and M.P. Vecchi, Science 220 (983), p A. Corana, M. Marchesi, C. Martini and S. Ridella, ACM Trans. Math. Software 3 (987), p 262 Page 83 / 42

84 Flow Chart of Magnetic Structure Determination Nuclear space group G 0 Peak positions propagation vector k Neutron diffraction experiment Little group G k Irreducible representations of G k Atom Position Basis vectors for symmetry related (equivalent) atoms Symmetry Calculation Neutron diffraction Intensity measurements Least squares / Reverse Monte Carlo Magnetic Structure(s) Mx, My, Mz Page 84 / 42

85 Magnetic Structure Group Theory Andrew S. Wills

86 Magnetic structures and their determination using Group Theory A. S. Wills Departement of Chemistry, Christopher-Ingold Laboratories University College London, 20 Gordon Street, London, WHH 0AJ. based on the JDN9 Ecole Magnétisme et Neutrons, May 2000, published in J. Phys. IV France, Pr9-33 (200). Introduction The determination of magnetic structures is a special area of condensed matter research. While being fundamental to the understanding of electronic structures and properties, it remains a subject that is treated with difficulty and is full of incorrect solutions. This article is based around two goals: The explanation of the different possible types of magnetic structure. The demonstration of how symmetry leads to their proper description, and can aid their solution. In content, the first part of this article is based on the practicalities of what an experimenter should know in order to understand and describe a magnetic structure. In the second part, symmetry arguments will be shown to reduce the otherwise arduous task of determining a magnetic structure, to the investigation of a handful of possible structures. a.s.wills@ucl.ac.uk Page 86 / 42

87 Figure : Some different types of magnetic structures. 2 Page 87 / 42

88 2 Basic crystallography 2. Nuclear crystal structures A nuclear crystal structure can be described in terms of lattice translations of a unit cell. If the unit cell contains only one atom it is said to be a primitive cell; if it contains several atoms it is said to be a non-primitive lattice. The atomic positions of an arbitrary atom in the lth unit cell is given by where and R tj = t + r j () t = n a + n 2 b + n 3 c (2) r j = xa + yb + zc (3) Here a, b, c are unit vectors of the nuclear cell defined according to the International Tables; n, n 2, n 3 are integers and x, y, z have values that are less than unity. 2.2 Reciprocal lattice In crystallography a useful and much used construction is the reciprocal lattice - this can be defined as: a = 2π v 0 b c (4) b = 2π v 0 c a (5) c = 2π a b (6) v 0 Where v 0 is the volume of the unit cell, v 0 = a (b c). A reciprocal lattice vector τ connects the origin to a given node in reciprocal space when h, k, and l are integer numbers. τ = ha + kb + lc (7) 3 Page 88 / 42

89 3 Propagation vector and its star 3. Description of moments Before we detail what a magnetic structure is, we must begin with a description of the magnetic moment itself. There are of course a variety of ways and coordinate systems that can be used to describe a magnetic moment, e.g. Cartesian, polar or crystallographic coordinates. While it is of course preferable to describe the particular properties of the final structure in the most useful system (e.g. a rotation of a moment away from an axis is best described in terms of an angle), in the general case it is easiest to describe a moment in terms of projections along the crystallographic axes. Rather than say that a moment is of unit length and makes an angle of 0 with the c-axis, we will simply say that the projection of the moments along the crystallographic axes can be described by a basis vector Ψ which has components along these axes. In this case, the basis vector is Ψ = (00). In fact, when the basis vector is real, it simply corresponds to the projection of the moment along the different crystallographic axes, and so : m j = Ψ j (8) Often, however, the projections of the moment are described not just by one basis vector, but by the summation of several (see Section 5) : Ψ j = ν C ν ψ ν (9) In this work we will use ψ ν to represent the ν components of Ψ j for a given propagation vector k. The values of Ψ j will be taken as being those of atom j in the zeroth unit cell (i.e. the crystallographic cell). 3.2 Formalism of a propagation vector k Magnetic structures can be described by the periodic repetition of a magnetic unit cell, just as crystal structures are described by translation of a nuclear unit cell. For convenience, rather than building a complete magnetic unit cell (which could contain thousands of magnetic atoms) we use a description based on the nuclear unit cell and a propagation vector, k, that describes the relation between moment orientations of equivalent magnetic atoms in different nuclear unit cells. This provides a simple and a general formalism for the description of a magnetic structure. 4 Page 89 / 42

90 We illustrate this for the moment distribution m j associated with the atom j of a magnetic structure. This can be Fourier expanded, whatever the nature of the ordering, according to: m j = k Ψ k j e 2πik t (0) That the summation is made over several wave vectors that are confined to the first Brillouin zone of the Bravais lattice of the nuclear cell is explained in detail in Section 3.4. If only one wave vector is involved, this simplifies to: m j = Ψ k j e 2πik t () This equation describes the translation properties in real space of the basis vector Ψ j, which at present we can think of as the projections of the magnetic moment along the a, b, c crystallographic axes with relation to theatomicsiteinthezeroth (nuclear) cell. At another atomic site (of the same type) in the crystal that is related by a lattice translation vector t, the projections of the moment on the 3 crystallographic axes are related to those in the nuclear cell by Equation. An example of this is shown in Figure 2. Here the magnetic unit cell is 2 times larger along the c-axis than the nuclear unit cell and the propagation vector is k=00. The moment in the zeroth 2 cell is described by the basis vector Ψ j =(0 0), that is to say the moment is pointing along the b-axis. When we move to the cell above (i.e. toasitethat is related by the translation vector t=(0 0 ) the moment is rotated by 80 and now points along the (0-0) direction. As we move up the structure we find that the moment turns by 80 for each nuclear cell translation until at t=(0 0 2) it is the same as in the zeroth cell. In this way, if we know the basis vector that describes the moment orientation in the zeroth cell and the propagation vector, we can use Equation to calculate the basis vector and moment orientation, of any equivalent atom in the crystal structure. 3.3 Stability of magnetic structures When the sample is cooled and condenses into a state with magnetic order, the magnetic structure that results must leave the Hamiltonian invariant to lattice translations, i.e. the magnetic Hamiltonian of different unit cells must be the same. The minimisation in the magnetic energy of the system results in three possible situations: Remember that a lattice translation vector in real space is given the symbol t, while one in reciprocal space has the symbol τ. 5 Page 90 / 42

91 exp[-2ik t exp[-2 i(0 0 ½) (004) exp[-2ik t exp[-2i(0 0 ½) (003) exp[-2ik t exp[-2i(0 0 ½) (002) c exp[-2ik t exp[-2i(0 0 ½) (00) a b exp[-2ik t exp[-2i(0 0 ½) (000) Figure 2: Description of translational properties with the propagation vector k. In this example the basis vector for the moment in the zeroth cell is Ψ =(00),k=00 and each plane corresponds to a lattice translation of t=00 2 one k vector is more favourable than the others and the system chooses a ground state configuration that is described by: m j = Ψ k j e 2πik t (2) This is the most common situation and most of this work will be devoted to single k structures. several k vectors of the star are involved. described by: The ground state is then m j = k Ψ k j e 2πik t (3) This is termed a multi-k configuration. One k vector and its harmonics are involved, e.g. k. The ground state 2 is then described by: m j = Ψ k j e 2πik t (4) harmonics of k 6 Page 9 / 42

92 If the transition involves several arms of the star of propagation vector k and their harmonic terms, we have the possibility of crossed harmonics that are sometimes referred to as intermodulations. 3.4 Star of the propagation vector -k We will now consider the effects of the space group, G 0, of our crystal structure on the propagation vector k. For ease we will separate each symmetry element g = {h, τ } into rotation and translation parts, these are h and τ respectively. The action of the rotation part h on the reciprocal vector k, results either in leaving k unchanged, or the generation of an unequivalent wave vector k : where, k = kh (5) k = k or k k (6) In the general case, a number of distinct propagation vectors will result from the operations of the rotational elements of the space group G 0 on the propagation vector k. The symmetry elements of G 0 may then be classed into cosets, where the first coset G k is made up of elements that do not change the vector k, the second coset (given the symbol g 2 ) transform it into the unequivalent vector k 2,andsoon.Ifg L represents the elements of the coset L, we can write this relation as k L = kg L (7) In this way, we find that the rotation elements of the space group G 0 gives rise to a set of unequivalent wave vectors. These we describe as being the star of the propagation vector k;[] each wave vector is an arm of the star (and example of a star is given in Figure 3). The number of arms, l k, that make up a star is of course equal to the number of cosets and cannot exceed the number of elements in G 0. If Crystal Electric Field (CEF) or higher-order exchange interactions (e.g. quadrupolar-type) are appropriate, several arms of a star can be involved in the structure, it is then said to be a multi-k structure (this notion will be expanded upon in Section 4). More often, the magnetic structure is the result of only the first k vector. For this reason, we will now focus on the rotation elements that leave k invariant. 7 Page 92 / 42

93 c * a * b * k 2 k 3 k 4 k Figure 3: The star of the propagation vector k=(x 0 0) in the tetragonal space group I4/mmm (point group D4h). 7 The arms of the star are: k =(x 0 0), k 2 =(0 -x 0), k 3 =(-x 00)andk 4 =(0 x 0) 3.5 The little group of the propagation vector -k The symmetry elements of G 0 that leave the k vector invariant are of particular importance in the determination of a magnetic structure. For this reason the elements of the first coset are given a special name- they make up the little group G k, and it is on these that all the Group Theory arguments that follow in Sections 7 and 8 are based. The little group will be discussed in greater detail in Section 8. 4 Multi-domain and multi-k structures While the majority of magnetic structures that we come across involve only a single propagation vector k, it is useful to see how the different types of propagation vectors can take part in a magnetic structure. Experimentally, these situations are revealed by the appearance of more than a single reflection around a reciprocal lattice point. 4. Multi-domain structures The first magnetic neutron diffraction pattern collected was that of MnO, published by Shull and Smart[2]. The period of the magnetic unit cell was found to be doubled along each of the cubic axes of the FCC structure, and so its volume is 8 times that of the crystallographic cell. We now know that the structure in fact involves domains that order according to the 4 different arms of the propagation vector. The four k vectors involved are: k = ( 2, 2 2), ( k 2 = 2, 2, ) ( k 3 = 2 2, 2, ) ( k 4 = 2 2, 2, ) 2 (8) 8 Page 93 / 42

94 Mn O k 3 c * a * b * k k 2 k 4 Figure 4: a) The magnetic motif of MnO made up of ferromagnetic planes of moments that are coupled antiferromagnetically. b) The star of k in reciprocal space is made up of the four propagation vectors related by the rotation elements of the space group G 0 : k = (,, ) 2 2 2, k2 = (,, ) 2 2 2, k3 = (,, ) and k4 = (,, ) Domains are found that correspond to each of these k-vectors. As there are domains that order according to vector k,otherstok 2,..., k 4, this is termed a multi-domain structure. Experimentally, different k domains will lead to different magnetic reflections, just as in multi-k structures. In fact, the diffraction patterns of multiple domain and multi-k structures are identical and it is impossible to distinguish them without the application of an external constraint that breaks the symmetry on a macroscopic scale, and favours the population of one k domain over another. 4.2 k and -k structures Structures that involve contributions from the two arms k and -k do not fall simply into the class of multi-k structures because, as we will show in Section 5, the requirement of a contribution from the -k arm can simply be the result of the form of the basis vectors, or the value of k. Typically, the contribution of these two components gives rise to modulated magnetic structures, e.g. sine and ellipse structures. 9 Page 94 / 42

95 4.3 Multi -k structures As we have already seen, multi-k structures can involve different arms of the star of the propagation vector k. This is a situation favoured by higher terms in the exchange Hamiltonian of the magnetic system. Also possible are structures that involve an accidental degeneracy between the stars of unrelated propagation vectors. A magnetic transition that involves several stars does not necessarily follow the Landau theory for a second-order transition (Sections 7.3 and 7.2), but if suitable degeneracies occur the resulting structure may still order under a single Irreducible Representation. 4.4 Structures that involve the harmonics of k Addition of components of the harmonics of k to a structure will lead to a squaring up of the modulation, that is to say the magnitudes of the moments on the atoms becomes equal. This situation can be driven by CEF effects that disfavour any reduction in the amplitude of magnetic moment, or an instability of the modulated structure because of the large entropy associated with it. This is exemplified by a sine structure, where decreasing temperature leads to the structure becoming unstable and may lead to a squaring up of the modulation of the moments. Examples of this are the metals Er and Tm (see Figure 5) where third, fifth and higher order harmonics progressively appear with decreasing temperature.[3] 5 Translation properties of magnetic structures Now that we will return to the situations that involve only a single propagation vector k, and perhaps its inverse -k. Wehavealreadyshownhowbasis vectors and propagation vectors can be used in the description of magnetic structures. In this Section we will examine the different types of magnetic structures demonstrated in Figure. In the general case, the k vector may refer to any point within or on the surface of the first Brillouin zone. This gives rise to two general classes of magnetic structures: Commensurate- the magnetic cell that is a simple multiple of the nuclear cell. It is in this group that are found the majority of known magnetic structures: simple ferromagnets, antiferromagnets and ferrimagnets. 0 Page 95 / 42

96 Figure 5: The magnetic structures of the heavy rare earth metals. Incommensurate- there is no simple relation between the structural and magnetic cells. It is important to note that these classifications describe only the propagation vector; the magnetic structure itself is the result of the propagation vector k and the basis vector Ψ j. It is the combination of both of these that gives rise to the different possible structures[, 4, 5, 6]. 5. Simple structures and Sine structures As we have seen, the translation properties of a magnetic structure may be described by: Let us now expand the exponential: m j = Ψ k j e 2πik t (9) m j = Ψ k j [cos( 2πk t)+isin( 2πk t)] (20) and consider various possibilities for the basis vector Ψ j and the propagation vector k. Page 96 / 42

97 5.. Ψ is real and the sine component is null The simplest situation occurs when Ψ k j is a real basis vector. The condition that m j is real requires that the sine component is zero- this occurs only for certain values of k. Equation20thenreducesto m j =Re(Ψ k j cos( 2πk t)) (2) As the sine component is null, the cosine component is necessarily of maximal magnitude and so translation to another unit cell results only in some rotation of the moment, and does not change its magnitude. This is the situation in many simple ferromagnetic, ferrimagnetic, and antiferromagnetic structures (examples are given in Figure a-f).) 5..2 Ψ is real and the sine component is non-zero If the basis vector is real and the sine component is non-zero, Equation 20 leads to a magnetic moment that is complex- an impossible situation as the magnetic moment is a real entity. We are therefore left with the problem of how to relate our complex basis vector to the projections of a real moment. This in fact turns out to be very simple: the moment here cannot be described by a single propagation vector, but rather is described by contributions from 2 propagation vectors. The second propagation vector that is required in order to describe the magnetic moment distribution is the propagation vector -k. The atomic vector for an atom in the nth cell related to that in the zeroth cell by translation t is then given by:[4] Where [4], m j = Ψ k j e 2πik t + Ψ k j e 2πik t, (22) Ψ k j = Ψ k j (23) Insertion of this relation into Equation 22 and expansion of the exponential leads to m j = 2Re(Ψ k j )cos( 2πk t)+2im(ψk j )sin( 2πk t) (24) As we are considering real basis vectors, the imaginary component in Equation 24 is zero and this reduces to m j = 2Re(Ψ k j )cos( 2πk t) (25) We therefore see that if the propagation vector k leads a non-zero sine component in Equation 20, the magnetic structure involves both the wave vectors 2 Page 97 / 42

98 k and -k. A non-zero sine component requires also that the magnitude of the moment changes with translation through the crystal. The resulting structure has a sine modulation and an example is shown in Figure h. 5.2 Helical structures 5.2. Ψ is complex and Re(Ψ)=Im(Ψ) A complex basis vector associated with the vector k requires also a contribution from the -k. Therefore, we begin again from Equation 24: m j = 2Re(Ψ k j )cos( 2πk t)+2im(ψk j )sin( 2πk t) (26) If the real and imaginary components of Ψ are equal we find that this simplifies to: m j = 2Re(Ψ k j )[cos( 2πk t)+sin( 2πk t)] (27) As the sine and cosine components define the points on a circle, the resulting structure is said to be a circular helix, i.e. one in which the magnitude of the moment is constant, but its orientation changes (Figure i) Ψ is complex and Re(Ψ) Im(Ψ) As the real and imaginary components are of different size, the equation m j = 2Re(Ψ k j )cos( 2πk t)+2im(ψk j )sin( 2πk t) (28) describes an ellipse rather than a circle. The resulting structure is referred to as an elliptical helix (Figure j). 5.3 Summary of structures and basis vectors In this Section we have shown that the class of a magnetic structure is the result of both the propagation vector and the form of the basis vectors involved. Sine structures and simple structures arise from real basis vectors, while helices involve complex basis vectors. The key equations are and m j = Ψ k j e 2πik t (29) m j = 2Re(Ψ k j )cos( 2πk t)+2im(ψk j )sin( 2πk t) (30) 3 Page 98 / 42

99 The calculation of these basis vectors will be detailed later in the section on Group Theory calculations (Section 8). 6 Location of magnetic reflections 6. k=0 ferromagnetic (ferri- antiferromagnetic) As the magnetic and crystallographic unit cells are of the same size, the magnetic reflections occur at the nodes of the nuclear reciprocal lattice and their intensities therefore add to those of the nuclear reflections. Assuming an unpolarised incident beam, the magnetic cross section[7, 8] is then given in barns by: ( ) dσ = M (Q) 2 δ(q τ ) (3) dω mag τ Where τ is a reciprocal space vector as defined by Equation 7 and M (Q) is the magnetic interaction vector (the component of the magnetic structure factor perpendicular to the scattering vector Q, in units of 0 2 cm): M (Q) = F M (Q), (32) where F M (Q) has units of Bohr magnetons. 6.2 k 0 antiferromagnetic- commensurate The example shown in Figure 6b is of the propagation vector k=( 00). As 2 the magnetic unit cell is 2 times larger in the a direction than the nuclear cell, the reflections will occur at half-integer positions ( hkl) k 0 antiferromagnetic- incommensurate We know that due to the form of Equation, contributions from the basis vectors of both k and -k are required. Reflections will therefore be at positions associated with both of these propagation vectors, and pairs of Bragg reflections will surround each reciprocal lattice point. As demonstrated in Figure 6c, magnetic reflections will be observed at:. For k Q = τ + k : reflection τ + or (hkl) + 4 Page 99 / 42

100 2. For -k Q = τ k : reflection τ or (hkl) 6.4 Multi-k Figure 6d demonstrates the diffraction pattern of a structure described by two incommensurate propagation vectors. We see that there is a pair of reflections for each propagation vector about the reciprocal lattice points. This magnetic pattern is exactly the same as that from a structure with two equally populated k domains and only the application of a suitable external constraint can allow the distinction of these situations. 6.5 Harmonics of k Contributions from the harmonics of k lead to the occurrence of reflections at positions that correspond to fractions of k. The example given in Figure 6 is of an incommensurate propagation vector and its harmonic 2 k. 7 Symmetry in magnetic structures 7. The little group G k and its irreducible representations As we have already stated, the little group G k that is made up of all the symmetry elements that leave k invariant, is a central concept in the symmetry analysis of magnetic structures. For a magnetic structure to be possible, it must be compatible with all of the symmetry operations of G k simultaneously. The set of matrices that describes how the moments transform under all of the operations of G k makes up a representation. It is useful to separate these representations into orthogonal Irreducible Representations[9] (IRs), just as we separate the vibrations of a molecule into normal modes. 7.2 Landau Theory and its application to magnetic phase transitions and structures The power and utility of Group Theory calculations with regards to the determination of magnetic structures comes from the Landau theory of a second-order phase transition. In the simplest of terms, this states that a 5 Page 00 / 42

101 a) k=0 ferromagnetism (ferri- or antiferromagnetism in non-primitive cell) d FM ( Q) Q - d ( ) M b * a * b) 2 k= 00 antiferromagnetism (commensurate propagation vector) c) k=kxky kz antiferromagnetism (incommensurate propagation vector.) Contributions from both k and -kare required. k Q= + k - k Q= - k d) multi-k: kx ky k z + R (kx ky k z) +... antiferromagnetism (incommensurate propagation vector.) e) k with harmonics antiferromagnetism (incommensurate propagation vector.) Figure 6: Cross-sections and graphs in reciprocal space for a variety of magnetic structure classes 6 Page 0 / 42

102 second-order transition can involve the build up of magnetic fluctuations that have the symmetry of only one Irreducible Representation (in this case an Irreducible Representation describes the symmetry properties of a magnetic moment under all the symmetry operations of the little group G k )[4, 6, 9]. Because of this, the resulting magnetic structure can be described by the basis vectors associated with only that Irreducible Representation and the basis vectors associated with the Irreducible Representations not involved in the transition are necessarily zero. This greatly limits the number of possible magnetic models and the number of parameters that are involved in their refinement Even in the cases where the transition is not second-order, nature is often kind to us and the structures that result are often the same as would be predicted for a second-order transition.the calculations detailed in Section 8 therefore continue to constitute a useful step in the determination, and description, of a magnetic structure. 7.3 Application to structures with several magnetic sites If the unit cell of interest has several magnetic sites we have to consider how they will behave. If there are two types of site, A and B, there are 3 limiting cases and we will consider each separately[4]: The two intra-site interactions are dominant: I A >I B >I AB. Here the coupling between the sites is small and so the sites behave independently. Each will therefore have its own ordering transition and no relation between the different Irreducible Representations involved is necessary. The inter-site interactions are dominant: I AB >I A >I B. The strong coupling between the sites leads to a single critical temperature. The basis vectors that are associated with both sites must belong to the same Irreducible Representation. This places a great restriction on the number of possible structures. One intra-site interaction is dominant: I A >I AB >I B. Upon cooling 2 distinct phase transitions will occur. The first involves the moments on the A sites. The inter-site coupling will lead to this structure polarising the B moments. These will then display the same magnetic structure as the A atoms. At a lower temperature, the B 7 Page 02 / 42

103 moments will undergo a symmetry-breaking transition and order cooperatively. The strong coupling between the A and B sites requires that these two orderings involve the same Irreducible Representation. As an example, let us consider a system where there are 4 possible Irreducible Representations: Site A: Γ +0Γ 2 +Γ 3 +Γ 4 Site B: Γ +Γ 2 +0Γ 3 +0Γ 4 We know that only non-zero Irreducible Representations (labelled Γ) can be responsible for a magnetic structure. We see immediately that not all the Irreducible Representations occur on the two sites, i.e. on site A, Γ 2 is not involved. If site A orders separately, the resulting structure will correspond to either that of Γ,Γ 3 or Γ 4, that is to say there are 3 possible magnetic models. Similarly site B could order according to Γ or Γ 2. If there is no coupling between the sites and each orders separately, there are no symmetry restrictions on the possible Irreducible Representations involved. The sites can therefore order according to any of their allowed Irreducible Representations. However, if the situation is such that both order together, the two sites must order under the same Irreducible Representation, and only the Irreducible Representation Γ can lead to a magnetic structure. The determination of the magnetic structure is therefore greatly simplified, as it can only involve the basis vectors associated with Γ. 8 Representational Analysis 8. Group Theory and magnetic structures In non-primitive cells we must also determine the relation between the different magnetic moments in the cell. This relation can be very difficult to derive and is often found by comparison with known magnetic structures, or by trial and error. Group Theory arguments allow us to calculate symmetryallowed relations between the moments and to greatly simplify this process. The results of these calculations are precisely the basis vectors, that we have been using to describe the magnetic structures. The technique that will be presented in this work involving the application of Group Theory to magnetic structures is termed Representational Analysis[0,, ]. The only pieces of information that are required for these calculations are the propagation vector k, the crystallographic space group 8 Page 03 / 42

104 and the atomic coordinates of the magnetic atoms before the magnetic phase transition. Rather than simply detailing the calculations involved, their application to an example problem will be used. Figure 7: The kagomé lattice. 8.2 Computer programs A number of computer programs exist that perform the calculations that make up magnetic symmetry analysis. Irreducible Representations can be calculated using KAREP[2], MODY[3], BASIREPS[4], and SARAh[5]. Basis vectors for the symmetry-allowed magnetic structures can be calculated using MODY, BASIREPS, or SARAh. All the Group Theory calculations and refinements presented here have been made using the program SARAh. 8.3 Example: of AgFe 3 (SO 4 ) 2 (OH) 6 with k = The jarosites are described in the space group R 3m (point group D3d 5 )and their crystal structure is displayed in Figure 8. All the calculations that follow will refer to the hexagonal non-primative setting of this space group, the symmetry elements of which are given in Table. As the cell is hexagonal there are three kagomé layers in the crystal structure and these have the stacking sequence...abc... The magnetic Fe 3+ ions make up a 2-dimensional geometry called a kagomé lattice (Figure 7). In the mineral argento-jarosite, AgFe 3 (SO 4 ) 2 (OH) 6, the exchange is antiferromagnetic and magnetic ordering with a propagation vector k =00 3 (with respect to the hexagonal axes) has 2 been found at low temperature.[6, 7, 8] In this section we will calculate the symmetry-allowed magnetic structures using Representational Analysis. These calculations are also detailed in Ref. [7]. 9 Page 04 / 42

105 Element number IT notation Jones symbol Rotation matrix 0 0 g { 000} x, y, z g2 { } ȳ, x y, z g3 {3 000} y x, x, z g4 {2 000} y, x, z g5 {2 000} x y, ȳ, z g6 {2 000} x, y x, z g7 { 000} x, ȳ, z g8 { } y, y x, z g9 { 3 000} x y, x, z g0 {m 000} ȳ, x, z g {m 000} y x, y, z g2 {m 000} x, x y, z Table : Symmetry elements of the space group R 3m. The notations used are of the International Tables, where the elements are separated into rotation and translation components, and the Jones faithful representations of the rotation parts. The latter corresponds to the vector formed from the operation of the rotation part of the element on (x, y, z). 20 Page 05 / 42

106 Figure 8: The jarosite crystal structure in the space group R 3m. 8.4 The group G k and its Irreducible Representations As we have already explained in Section 3.4, for a given propagation vector k, some of the operators of the space group G 0, g = {h τ }, leave it invariant while others transform it into an equivalent vector that differs by some arbitrary translation of the reciprocal lattice, τ, according to: kh = k + τ (33) This set of elements makes up the so-called little group, G k, which is a subgroup of G 0. The Irreducible Representations of this little group are given by the symbol Γ ν,whereν is the label of the irreducible representation, and the matrix that corresponds to the symmetry element g is labelled by d k ν(g) Looking at the example of AgFe 3 (SO 4 ) 2 (OH) 6 with k =00 3, we find that 2 the little group contains all of the 2 symmetry operators of the space group R 3m. The Irreducible Representations of these are given in Tables 2 and 3. One sees immediately that the second-order representations Γ 5 and Γ 6 have the same elements for symmetry operations -6 and are related by a factor of (-) for the operations 7-2. These Irreducible Representations may be verified against tabulated val- 2 Page 06 / 42

107 g g2 g3 g4 g5 g6 g7 g8 g9 g0 g g2 Γ Γ Γ Γ Table 2: First-order Irreducible Representations for the group D 5 3d for the vector k= Γ 5, Γ Γ Γ g g2 g3 g4 g5 g6 ɛ 0 ɛ ɛ 2 0 ɛ 2 0 ɛ 0 ɛ 0 0 ɛ ɛ 2 0 g7 g8 g9 g0 g g2 ɛ 0 ɛ ɛ 2 0 ɛ 0 ɛ 2 0 ɛ 0 ɛ 0 ɛ 2 0 ɛ 0 ɛ ɛ 2 0 ɛ 0 ɛ 2 0 ɛ 0 ɛ 0 ɛ 2 0 Table 3: Second-order Irreducible Representations for the group D3d 5 vector k= ɛ=exp(- 2π ). 2 3 for the ues of the projective (or loaded ) representations, d pr ν, given in works such as Bradley and Cracknell[9] and Kovalev[9]. The tabulated representations are given for the various point group symmetries and can be converted into the Irreducible Representations of the little group G k of the propagation vector k by multiplicating them with a phase factor: d ν = d pr ν e 2πk τ (34) Where τ represents the translation part of the symmetry operator to which d ν is associated. 8.5 Effect of symmetry element on a moment bearing atom The effect of a symmetry element is two-fold: it will act to change the position of an atom, and reorientate the magnetic moment, e.g. atom moves to the position of atom 2, and its moment is reversed. The combination of these two results are described by the magnetic representation, Γ. We will examine these two effects separately: 22 Page 07 / 42

108 8.5. Effect of symmetry element on atom positions: the permutation representation A symmetry operator g = {h τ } acts on both the position r j of the atom and on the components α of the axial vector that describes the moment. The operation that sends r j in the zeroth cell to r i in the pth cell can be symbolically stated as g(j0) (ia p ) (35) In other terms, the effect of a symmetry operation g is to permute the column matrix of atom labels, P: g(p) P (36) This operation is governed by a permutation representation, Γ perm, which has matrices of order N A,whereN A is the number of equivalent positions of the crystallographic site. It is important to note that when a symmetry operation results in an atomic position that is outside the zeroth cell, a phase factor must be included that relates the generated position to that in the zeroth cell. This phase is simply given by: θ = 2πk T (37) Where T is the translation vector, that relates the original and generated atoms. As an example, from Table 4 we see that the permutation equation for the atoms of the three Bravais sublattices under the g={ } operation is: 2 exp(θ a ) 3 exp(θ b ) exp(θ c ) = Γ perm 2 3 (38) Where the atomic positions follow the labelling: =( ), 2=( 0 ), =(0 ). For the operation 2 2 g={3+ 000}, θ a = θ b = θ c =0fork= The permutation representation is therefore given by 2 Γ {3+ 000} perm = (39) 23 Page 08 / 42

109 The character of this representation, χ perm, for each symmetry operator is then simply the sum of the phases θ(g) for the atoms that are transformed into an equivalent atom under a symmetry operation, and so for both the propagation vectors, χ {3+ 000} perm = Effect of symmetry element on moment vectors: the axial vector representation The second effect of this symmetry operation is to transform the spin components with index α, (α = x, y, z) of the reference spin j into the index α of the atom at r i.[0,, ] These transformations are described by the axial vector representation, Ṽ, the character of which is given by χ h Ṽ = a=b R h abdet(h), (40) Where R h ab refers to a specific element a,b of the rotation matrix h, anddet(h) represents the determinant of the rotation matrix R h,and has the value of + for a proper and - for an improper rotation. This is exemplified for the 3 + rotation, where the operation of h(3 + ) on the moment vector M=(m x m y m z )gives: R(3 + ) M = det(h) = m y m x m y m z m x m y m z (4) (42) As 3 + is a proper rotation, det(3 + ) = and the character of Ṽ for h(3+ )is therefore χ 3+ Ṽ = Magnetic representation As we have already stated, the magnetic representation, Γ, describes both the result of the symmetry operation on the atomic positions, and on the axial vectors that describe the magnetic moments. As these effects are independent, the magnetic representation is given by their direct product[0,, ]: Γ=Ṽ Γ Perm (43) Or, in terms of the matrices for the representations themselves 24 Page 09 / 42

110 Atoms Axial vector components g = {h τ } 2 3 χ perm m x m y m z χṽ { 000} m x m y m z 3 { } m y m x -m y m z 0 {3 000} m x +m y -m x m z 0 {2 000} m y m x -m z - {2 000} m x -m y -m y -m z - {2 000} m x -m x +m y -m z - { 000} m x m y m z -3 { } m y m x -m y m z 0 { 3 000} m x +m y -m x m z 0 {m 000} 3 2 m y m x -m z - {m 000} 3 2 m x -m y -m y -m z - {m 000} 2 3 -m x -m x +m y -m z - Table 4: The permutation of B 3+ atoms (at position 9d) and the transformation of the axial components of the moment under the different symmetry operators of the R 3m space group (point group D3d 5 )fork = 003. The 2 characters of the representations Γ perm and Ṽ are given. D(h,τ Γ h ) = DṼ(h) D Γ Perm (h,τ h ) (44) The characters of these representations are related according to: χ Γ = χṽ χ perm (45) 8.7 Reduction of the Representation Γ The magnetic representation for a particular site can be decomposed into contributions from the Irreducible Representations of the little group: Γ= ν n ν Γ ν (46) where n ν is the number of times the irreducible representation Γ ν appears in the magnetic representation Γ. n ν is given by: n ν = χ Γ (h)χ Γ n(g k ) ν (h) h G k (47) Here, χ Γ is the character of the magnetic space group and χ Γ ν is the complex conjugate of the character of the irreducible representation with index ν. 25 Page 0 / 42

111 8.8 Calculation of the basis vectors Ψ The basis vectors, ψ n, that transform according to the µ dimensional irreducible representation Γ (µ) ν are projected out of the representation matrix D ν using a series of test functions φ β,whereφ = (00), φ 2 = (00), and φ 3 = (00). This is carried out by the projection operator formula: ψ iλ n = D λ ν (g)δ i,gi e 2πik (r gi r i ) det(h)r h φ β, (48) g G k The summation is over the symmetry elements of the little group G k. ψ is a spin component that we represent by a column matrix ψ(r). δ i,gi is unity if the atoms i and giare equivalent positions of the crystallographic site that are related by a primitive lattice translation, i.e. they are of the same sublattice of the Wyckoff site. Equation 48 is applied sequentially to each element λ of the matrix D ν, for each equivalent position i of the crystallographic site. The row of the matrices D ν is fixed during the examination of a given IR. In our calculations the µ elements are those that correspond to the first row of the matrix of D ν. As for each element, labelled λ =...µ, three components β are projected out, there are in total 3µ projected components. Of these, the number of non-zero unique projected components for a representation is of course the same as calculated using Equation (47). 8.9 Refinement of basis vectors mixing coefficients Any linear combination of basis vectors within one representation is necessarily a symmetry-allowed basis vector. The atomic moment on a particular atom,m j, is therefore most generally given by the sum of the basis vectors for a particular irreducible representation: m j = ν C ν ψ ν, (49) where C ν is the mixing coefficient of the basis vector ν. In refining the orientation of an atomic moment, we are in effect refining the mixing coefficients C ν of the basis vectors within the irreducible representation being examined. The number of variables in our refinement is simply the number of unique basis vectors that transform according to a given representation, i.e. n ν µ. 8.0 Refinement of complex basis vectors The refinement of the mixing coefficients that relate complex basis vectors will be dealt with in detail, to demonstrate how an ordered array of magnetic 26 Page / 42

112 moments, which are necessarily real entities, can be described by complex basis vectors. 8. Decomposition of the magnetic representation and the basis vectors of AgFe 3 (SO 4 ) 2 (OH) 6 In the hexagonal setting the magnetic Fe 3+ ions are found on the 9d sites. For these sites the decomposition of the magnetic representation according to Equation 47 is: Γ=0Γ () +2Γ () 2 +0Γ () 3 +Γ (2) 4 +3Γ (2) 5 +0Γ (2) 6 (50) The Landau theory of a second-order phase transition, requires that only one representation is involved, and so for this k there are only three possible magnetic structures. These correspond to representations Γ 2,Γ 4 and Γ 5. The basis vectors for these representations calculated according to Equation 48 are given in Table 5. The atomic sites are labelled following the convention given in Section The basis vectors have varied forms and we will now explain in detail the types of magnetic structures that they correspond to. IR b-v Atom Atom 2 Atom 3 m x m y m z m x m y m z m x m y m z Γ 2 ψ ψ Γ 4 ψ Γ 5 ψ ψ ψ ψ ψ ψ Table 5: The basis vectors of the Irreducible Group Representations of the space group R 3m (point group D3d 5 ) appearing in the magnetic representation with k = Representations Γ 2 and Γ 4 are one dimensional. They therefore correspond to simple magnetic structures in which the atomic moments are orientated along particular crystallographic axes. It is noteworthy that both ψ and ψ 2 correspond to 20 spin structures, with the total spin on any given 27 Page 2 / 42

113 triangle plaquette being is i =0. While, the two spin structures are in fact related by a global rotation of spins, the two representations differ in that Γ 2 allows the introduction of an out-of-plane component, which corresponds to ψ 2. The combination of the 2 basis functions ψ and ψ 2 creates a so-called umbrella structure (of the type shown in Figure f) in which the degree of out-of-plane canting is a refinement variable. Representation Γ 5 is two dimensional and is repeated 3 times. It therefore corresponds to a 6 basis vector magnetic structure. As the general solution involves any linear combination of these 6 basis vectors we can not ascribe to this representation a simple structure. We do note however, that there are relations between the basis vectors and these will simplify the refinement of the mixing coefficients: ψ 4 =ψ 8, ψ 5 =ψ 7 and ψ 6 =-ψ 9. For pedagogical reasons we will ignore these relations and continue as if Γ 5 involved six untelated complex basis vectors. In this case, as the atomic spins are real entities, it is necessary to introduce the corresponding basis vectors of the propagation vector -k in order to make the summation of the two components real (Section 5). A description of the translation properties of this structures begins as normal from: m j = Ψ k j e 2πik t + Ψ k j e 2πik t, (5) However, as for k =00 3 the vectors k and -k are equivalent, we have k = k 2 and so and e 2πik t = e 2πik t (52) m j =(Ψ k j + Ψ k j )e 2πik t (53) A further simplification arises from the fact that the addition of the -k contribution corresponds to the addition of the conjugate of the basis vector of k, i.e. We therefore obtain Ψ k j = Ψ k j (54) m j = 2Re(Ψ k j )[cos(2πk t n )+isin(2πk t)] (55) For both k =00 3 the sine component vanishes under the centring translations of the non-primitive cell, or integer translations of the crystallographic 2 cell, and so Equation 55 reduces to 28 Page 3 / 42

114 m j =2Re(Ψ k j )cos(2πk t) (56) As in general, the basis vectors associated with Γ 5 that are to be tested against the magnetic structure are complex, when considering the translational properties of the magnetic moments it is sufficient just to add their complex conjugate in order to arrive at real values for the atomic moments. In these two cases this leads to Equation 56. The astute reader will have noted that while this procedure began with six basis vectors, it finished with only three: we have apparently halved the dimensionality of the basis vector space. In order to determine the remaining three basis vectors we must look at the imaginary part of ψ in Equation 53. This is done by first multiplying the basis vector ψ by the imaginary number i. Equation 53 would then read m j =(iψ k j + iψ k j )e 2πik t (57) and Equation 56 would become m j =2Im(Ψ k j )cos(2πk t) (58) As the basis vectors on Γ 5 are themselves related by complex conjugation, we do not need in fact need to resort to this addition of the components associated with -k. Instead, we can simply equate the mixing coeffients of the conjugate pairs, i.e., C(ψ 4 )=C(ψ 8),etc.. There is an alternative way of dealing with complex basis vectors. Instead of combinining the complex basis vectors with their conjugates as described above, we can use the properties of irreducible representations. In some cases 2 unitary rotations can be applied to the Irredicible Representations that make them real. These two techniques are equivalent. 8.2 Refinement of the magnetic structure of AgFe 3 (SO 4 ) 2 (OD) 6 The collected neutron diffraction data were found to be compatible only with a magnetic structure described by the representation Γ 4. Figure 9 displays the value of χ 2 as a function of the mixing coefficients C(ψ )andc(ψ 2 ); the only refinement variable in the least-squares matrix was the magnitude of the magnetic moment. In all cases the sum of the mixing coefficients was adjusted to be unity, and a trivial factor was then used to separately scale the magnitudes of moments described. The best value of χ 2 corresponds to the 2 this criteria for this transformation is that the Coirreducible Representation (CIR) derived from the Irreducible Representation is real 29 Page 4 / 42

115 C( ) C( 2 ) Figure 9: χ 2 as a function of the basis vector coefficients C(ψ )andc(ψ 2 ) during the refinement of the magnetic structure of AgFe 3 (SO 4 ) 2 (OD) 6 at.5k. coefficients C(ψ )=0.99 (5) and C(ψ 2 )=0.0 (5), that is to say the refined structure is coplanar and the contribution from out-of-plane canting is zero within the error of these data. The final refined profile is presented in Figure 0 and the final magnetic structure in Figure. 8.3 Discussion of the magnetic structure of AgFe 3 (SO 4 ) 2 (OD) 6 The magnetic structure that is refined is a triangular structure, that is to say the neighbouring moments are related by 20. This is what we would naïvely expect for a triangular array of antiferromagnetically coupled spins. The Group Theory arguments we have used indicate that only particular orientations are possible for this configuration. As we will show in the practicals that accompany this course, the out-of-plane component that is only allowed in Γ 2 is important at higher temperatures and leads to the formation of an umbrella structure. 9 Summary of course As stated in the introduction, this course was intended to explain how to describe a magnetic structure in terms of a propagation vector and some of its associated basis vectors. The examples given show how the different possible types of magnetic structure lead directly from this description. The second part of this text has been devoted to magnetic symmetry analysis- 30 Page 5 / 42

116 Figure 0: Experimental and calculated diffraction patterns for AgFe 3 (SO 4 ) 2 (OD) 6 at.5k. Magnetic and crystallographic reflections are indicated by the upper and lower tick marks respectively. Figure : Magnetic structure of AgFe 3 (SO 4 ) 2 (OD) 6. 3 Page 6 / 42

117 Representational Analysis. The Group Theory calculations that this involves are tedious, but now computer programs exist that perform these calculations in seconds. The basis vectors that result simplify greatly the processes of finding a magnetic structure, and can facilitate their correct description. 0 Further reading Much inspiration, of varying levels, has been taken from a number of works on magnetic structures and magnetic symmetry analysis. For a first step into these subjects the References [, 4, 5, 6, 7] are particular suitable. Reference [7] presents a clear introduction into the technical aspects of magnetic neutron scattering. Acknowledgements ASW would like to thank the Marie-Curie project of the EC for support and the Grenoble school of magnetism for their tuition, encouragement, and patience. References [] Yu.A. Izymov, V.E. Naish and R.P. Ozerov, Neutron Diffraction of Magnetic Materials, Consultants Bureau, New York, 99. [2] C.G. Shull and J.S. Smart, Phys. Rev. 76, 256 (949). [3] D. Gignoux, J.C. Gomez Sal, J. Magn. Magn. Mat., 203 (976). [4] J. Rossat-Mignod in: Methods in Experimental Physics, ed. K. Sköld and D.L. Price (Academic Press, 987). J. Rossat-Mignod in Systematics and the Propeties of the Lanthanides, Ed. S.P. Sinha, D. Reidel Publishing (983). [5] P.J. Brown, Physica B 37, 3 (986). [6] J. Schweizer, in: Hercules- Neutron and Synchrotron Radiation for Condensed Matter Research, Springer Verlag, Berlin (994). [7] F. Tasset, Recent developments in neutron scattering Proceedings of Les Houches École de Physique Page 7 / 42

118 [8] F. Tasset, Structures magnétiques et polarimétrie neutronique Proceedings of l Ecole Magnétisme et Neutrons de la Société Française de la Neutronique, a French Translation of Ref. [7] to appear in J. de Physique (200). [9] C.J. Bradley and A.P. Cracknell, The Mathematical Theory of Symmetry in Solids, Clarendon Press (Oxford, 972). [0] E.F. Bertaut, J. Appl. Phys. 33, 38 (962). E.F. Bertaut, Acta. Cryst. A24, 27 (968) E.F. Bertaut, J. de Physique, Colloque C, 462 (97). E.F. Bertaut, J. Magn. Magn. Mat. 24, 267 (98). [] Yu.A. Izyumov and V.E. Naish, J. Magn, Magn. Mat.2, 239 (979); Yu.A. Izyumov and V.E. Naish, J. Magn, Magn. Mat.2, 249 (979); Yu.A. Izyumov and V.E. Naish, J. Magn, Magn. Mat.3, 267 (979). [2] E.R. Hovestreydt, M.I. Aroyo and H. Wondratschek, J. Appl. Cryst. 25, 544 (992). [3] P. Czapnik and W. Sikora, unpublished work. [4] J. Rodriguez-Carvajal, unpublished work. [5] A.S. Wills, Physica B (2000), 680. Program available from ftp://ftp.ill.fr/pub/dif/sarah/ [6] A.S. Wills, A. Harrison, C. Ritter and R.I. Smith, Phys. Rev. B 6, 656 (2000). [7] A.S. Wills, Phys. Rev. B, 63, (200). [8] A.S. Wills, Proceedings of Highly Frustrated Magnetism 2000 Conference, to appear in Can. J. Phys. (200). [9] O.V. Kovalev, Irreducibe Representations of the Space Groups, Gordon and Breach (New York, 96). 33 Page 8 / 42

119 Proceedings on the International Conference on Neutron Scattering, Munich 200. (PROOF COPY) Applied Physics A Materials Science & Processing Symmetry and magnetic structure determination: developments in refinement techniques and examples A.S. Wills Institut Laue-Langevin, 6 Rue Jules Horowitz, BP 56, Grenoble Cedex 9, France Received: 2 July 200/Accepted: 24 October 200 Springer-Verlag 2002 Abstract. Group-theory techniques can aid greatly the determination of magnetic structures. The integration of their calculations into new and existing refinement programs is an ongoing development that will simplify and make more rigorous the analysis of experimental data. This paper presents an overview of the practical application of symmetry analysis to the determination of magnetic structures. Details are given of the different programs that perform these calculations and how refinements can be carried out using their results. Examples are presented that show how such analysis can be important in the interpretation of magnetic diffraction data, and to our reasoning of the causes for the observed ordering. PACS: b; z Despite immense technical progress since the first magnetic neutron diffraction experiments of Shull and Smart [], the determination of magnetic structures remains a subject that is typically limited by the data-analysis strategy: structures are generally determined by intuition or simple trial and error refinement. As a consequence, the literature is full of incorrect magnetic structures and incomplete refinements. While group-theory techniques can be applied to limit the number of trial structures, or to determine along which directions the spin components can lie, their calculations when carried out by hand are arduous. This has led to their being applied only when a problem warrants their use or, more commonly, when it is sufficiently close to an example already in the literature. Recently, a number of computer programs have been developed that allow the unspecialised user to perform these calculations automatically. Their integration with common refinement codes allow for the first time the simple and rigorous examination of which symmetry-allowed magnetic structures are compatible with collected data. In this article a brief overview is made of the practical application of using group theory to aid the determination of a.s.wills@ill.fr magnetic structures from both powder and single-crystal samples. Examples are given that demonstrate the importance of symmetry information for the correct analysis of magnetic diffraction data, and concomitantly to the understanding of the physical reasons for the formation of a long-range magnetic order. Representational analysis calculations The calculations can be separated into two parts. The first is a grouping according to symmetry of the possible magnetic structures that are compatible with both the space group of the crystal structure and the propagation vector k of the magnetic ordering. The second part involves the application of Landau theory as a tool to simplify which of these are possible as a result of a continuous second-order phase transition. While the grouping and the labelling of the different magnetic structures by their symmetry properties is completely general, the assumptions made that involve the Landau theory are subject to its limitations. The application of group theory to the determination of magnetic structures is termed representational analysis [2 6] and is based on the calculation of the Fourier components of an ordered magnetic structure that are compatible with the symmetry of the crystal space group before the phase transition and the propagation vector. The first step in the analysis is the identification of the propagationvectork associated with the phase transition, and which space-group symmetry operations leave it invariant. These operations form the little group G k. The symmetry elements of G k and the value of k are then used to determine the different irreducible representations (IRs) of G k.the different basis vectors (Fourier components of the magnetic structure), BVs, that are projected out from an irreducible representation define a basis-vector space that may be termed a symmetry-allowed model. The different IRs define orthogonal basis-vector spaces that can be used to conveniently classify the different possible magnetic structures. Page 9 / 42

120 2 2 Application of Landau theory The Landau theory of a second-order phase transition requires that the Hamiltonian of the system is invariant under the symmetry operations of G k. This leads to the requirement that for a second-order phase transition an ordered structure can be the result of only a single IR becoming critical. This typically reduces the number of trial structures and the variables that each involve. When the assumptions of Landau theory are not valid, for instance when the Hamiltonian possesses odd-powered terms, the mixing of components from different IRs becomes possible. Continuing with this logic, the observation of a magnetic structure that involves different IRs is suggestive of either successive ordering transitions for each IR, or additional terms in the magnetic Hamiltonian that relax the single- IR rule, e.g. crystal-field terms at sites with certain point symmetries. 3 Programs that perform these calculations While tabulated values of the IRs of the space groups have existed for many years [7, 8], they are prone to inaccuracies and their laborious use in calculations carried out by hand is perhaps the major reason for their restricted use. Preferable to many are the programs that have been written that calculate these IRs, or use files of tabulated values such as: KAREP [0] (calculated), MODY [] (unverified tabulated values), BasiReps [2] (based on KAREP), and SARAh [3] (with the choice of KAREP-based and computer-verified tabulated values). MODY, BasiReps, and SARAh also use these values to calculate directly the basis vectors associated with the different IRs, and so allow the rapid and simple calculation of the possible symmetry-allowed structures. 4 Refinement using the results of symmetry analysis The simplest and most general mathematical description of a magnetic structure is in terms of Fourier components: the basis vectors that result from the group theory. The panoply of different possible commensurate and modulated incommensurate structures can be simply understood in terms of the form of the basis vector(s) for a site and the value(s) of k. To simplify the refinement process, SARAh and BasiReps have been written to integrate with the standard refinement codes FullProf 2000 [4] (BasiReps and SARAh) and GSAS [5] (SARAh). The applicability of the technique is now limited principally by the choice of refinement codes 2. The reduction in data due to the magnetic form factor and the observation of only the component of the magnetisation perpendicular to the scattering vector result in instabilities and limitations when using conventional least-squares The exception to this is likely to be those presented in [9] as these have been subject to computer verification. 2 Perhaps the most important restriction arises from the absence of a propagation vector in GSAS it is consequently limited to only simple commensurate structures. refinements. These can be overcome by the reverse-monte Carlo-based algorithms which allow the automatic exploration of the degrees of freedom associated with a given magnetic structure and the identification of additional minima in the refinement [3]. In more complex cases the technique of simulated annealing can be employed: this uses a decreasing criterion to allow the more controlled evolution of the system that is required to bypass the false minima commonly associated with larger numbers of variables. 5 Ongoing development of refinement codes Due to the difficulties in magnetic structure refinement being greatest for data from powders, development of these new magnetic structure refinement codes has concentrated on their analysis. As part of a collaboration between the Institut Laue- Langevin, the Commissariat à lénergie Atomique, and the Laboratoire Léon Brillouin we are at present working on the extension of the FullProf package not only towards the refinement of unpolarised and polarised neutron-diffraction data collected from single crystals, but also towards data collected by the technique of spherical neutron polarimetry. After these developments the data collected by any technique, from conventional powder diffraction to even the most complex collection techniques, will be refinable in terms of symmetrygenerated basis vectors. 6 Canted antiferromagnetism in M 2 [Ni(CN 2 )] (where M = Mn and Fe) The first example of the application of these techniques is taken from the M 2 [NiCN 2 ] (where M = Mn and Fe) molecular solids [6]. These binary metal dicyanide molecular materials crystallise in the space group Pnnm. Below secondorder transitions at T 6 K for the Mn and T 9 K for the Fe compounds, these display long-range magnetic order with the propagation vector k = (000). Symmetry analysis of the magnetic M atom at the 2a position indicates that there are four symmetry-allowed models; these correspond to the IRs Γ, Γ 3, Γ 5,andΓ 7 in the notation of Kovalev [8, 9] and their basis vectors are given in Table. While data collected using powder neutron diffraction can be well fitted by a simple model of antiparallel spins (M = M2) that were free to rotate in the ab plane, this spin structure is not allowed by symmetry. Investigation of the symmetry-allowed models found that the data could only be well fitted by Γ 5. IR BV M M2 Γ ψ (00) (00 ) Γ 3 ψ 2 (00) (00) ψ 3 (0 0) (0 0) Γ 5 ψ 4 ( 0 0) ( 00) ψ 5 (00) (00) Γ 7 ψ 6 (00) (00) Table. Magnetic basis vectors, BVs, for the 2a site of the space group Pnmm with the propagation vector k = (0 0 0). M and M2 are the atomic positions (0 0 0) and ( ) Page 20 / 42 MS ID: APA59 2 March :30 CET

121 3 Inspection of the associated basis vectors (ψ 4 and ψ 5 ) shows that while the moments are antiferromagnetically aligned along a, an uncompensated magnetisation can exist along b. The presence of such a ferromagnetic spin canting has been confirmed by dc susceptibility data. 7 Rare-earth nickel borocarbides Very recently, symmetry analysis has provided important new information about the variety of magnetic orderings that are observed in the rare-earth nickel borocarbides RNi 2 B 2 C(R= Gd Lu, Y) [7]. In these materials Fermi surface nesting effects propagated via the RKKY interactions create a strong tendency for these materials to order magnetically with the propagation vector k = ( ). Despite this, a large number of different magnetic structures are observed for the series. The key to understanding the magnetism of these materials was the single-ion anisotropies of the rare earths. These are typically well defined and possess a characteristic energy scale that is far greater than that of the exchange interactions. Their effect is to force the magnetisation to point along specific crystallographic directions. Symmetry analysis of the different magnetic structures that are possible for the propagation vector k = ( ) in this system indicated that when the single-ion effects were incompatible with the symmetry-allowed directions for this propagation vector, the system orders according to another propagation vector that does allow the single-ion effects of the rare earth in question to be satisfied. 8 The jarosites The jarosites (AFe 3 (SO 4 ) 2 (OD) 6,whereA= Na +,K +,Rb +, Ag +,ND + 4, 2 Pb+2 ), are the most studied examples of kagomé antiferromagnets. They have been the object of much scrutiny as the magnetic sublattice makes up a geometry of vertexsharing triangles. This results in their having an infinite number of classical ground states in the presence of only nearestneighbour antiferromagnetic exchange interactions. Furtherneighbour interactions can raise this degeneracy and cause one particular ordered spin configuration to be favoured from the degenerate manifold. Unfortunately, the experimental determination of which occurs is strongly hindered by an ignorance of the particular degeneracy-breaking interaction. Not only did symmetry analysis provide a particularly effective tool for the reduction in the number of trial structures, but it also helped to understand them in terms of the different terms in the exchange Hamiltonian [8]. 9 Conclusion The tools are now available that allow the unspecialised researcher to use symmetry analysis to make simpler and more rigorous the refinement of magnetic neutron diffraction data. As demonstrated here, their application to even the simplest structures is important, as physically unreasonable models still often fit experimental data. References. C.G. Shull, J.S. Smart: Phys. Rev. 76, 256 (949) 2. E.F. Bertaut: J. Appl. Phys. 33, 38 (962) 3. E.F. Bertaut: Acta Crystallogr. A 24, 27 (968) 4. E.F. Bertaut: J. Phys. Colloq. C, 462 (97) 5. E.F. Bertaut: J. Magn. Magn. Mater. 24, 267 (98) 6. Y.A. Izyumov, V.E. Naish, R.P. Ozerov: Neutron Diffraction of Magnetic Materials (Consultants Bureau, New York 99) 7. C.J. Bradley, A.P. Cracknell: The Mathematical Theory of Symmetry in Solids (Clarendon, Oxford 972) 8. O.V. Kovalev: Irreducible Representations of the Space Groups (Gordon and Breach, New York 965) 9. O.V. Kovalev: Representations of the Crystallographic Space Groups, 2nd edn. (Gordon and Breach, Switzerland, 993) 0. E.R. Hovestreydt, M.I. Aroyo, H. Wondratschek: J. Appl. Crystallogr. 25, 544 (992). P. Czapnik, W. Sikora: program available from 2. J. Rodríguez-Carvajal: program available from juan@llb.saclay.cea.fr 3. A.S. Wills: Physica B 276, 680 (2000). Program available from ftp://ftp.ill.fr/pub/dif/sarah/ 4. J. Rodríguez-Carvajal: Physica B 92, 55 (993) 5. A.C. Larsen, R.B. von Dreele: program available from ftp://ftp.lanl.gov/public/gsas/ 6. A. Lappas, A.S. Wills, M.A. Green, M. Kurmoo, K. Prassides: Phys. Rev. B (2002), submitted 7. A.S. Wills, C. Detlefs, P.C. Canfield: Eur. Phys. J. B (200) submitted 8. A.S. Wills: Phys. Rev. B 63, (200) CE b Please quote city/town instead of country. CE c Any update? Page 2 / 42 Editor s or typesetter s annotations (will be removed before the final TEX run) MS ID: APA59 2 March :30 CET

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123 Page 23 / 42 Magnetic Refinements in GSAS Paolo G. Radaelli

124 GSAS magnetic options The approach of GSAS to magnetic structures is loosely based on Shubnikov groups. However, for each space group, not all Shubnicov groups generated from it are possible. The only possible ones are those corresponding to subgroups of index 2 of types I and IIa. In other words, the conventional unit cell must be in common between the parent group and the subgroup. In GSAS there is a straight implementation of the OG formalism, where primed operators (or lattices) correspond to red operators. Alternatively, one can always generate an additional magnetic phase with appropriate constraints. GSAS magnetic entries Phase: in the phase menu (keystrokes k-p-p), one has the option of selecting (m) whether the phase is nuclear, nuclear and magnetic or purely magnetic (a, b, c, respectively). Form factor: in the form factor editing menu (k-p-f) there is an option (m) to edit magnetic form factors. One can use the default values (warning! They are different for different oxidation states) or input user values (see ITC, volume C). Atoms: in the atom editing menu (k-l-a) there is an option (m) to assign magnetic moments to individual atoms. Within that menu, there is an option (s) to prime the group generators. GSAS automatically determines if the magnetic point group of the site is admissible, and, if so, for which spin directions. One can change colours with the c option. Once out of the s menu, one can change the spin components with the m option. Page 24 / 42

125 CAF (A ) FM(c) FM(ab) AFM(A) AFM(G) Mn +3 x in La 2-2x Sr +2x Mn 2 O 7 Mn +4 Page 25 / 42

126 Notes on the Layered Manganite example The manganite site is 4e [4mm]. Of the magnetic subgroups of [4mm], the only admissible one is 4m m. Consequently, the only possible magnetic space groups are * I4/mm m I4/m m m, I P 4/mm m and I P 4/m m m. Note that the first one is a ferromagnetic group. An immediate consequence of the site symmetry of the Mn site is that the spin has to be directed along the 4-fold axis. There are therefore only 4 magnetic structures generated with the Shubnikov approach. The layers are always FM, with the intra- and inter-bilayer coupling being FM or AFM. Note the significant number of magnetic structures which are observed, but cannot be generated with the Shubnikov approach. Page 26 / 42

127 Magnetic refinements - multi-phase approach Should the Shubnikov approach be insufficient to describe the magnetic structure, one can resort to introducing a second purely magnetic phase with appropriate constraints but lower symmetry This enables one to deal with any kind of commensurate structure, including the representation analysis. Here are a few tips:. If the magnetic phase has the same conventional cell as the nuclear one, the lattice and phase fraction constraints are straightforward. 2. If the magnetic cell is larger than the nuclear one, one has to remember that the phase fraction is proportional to the number of unit cells in the sample. So, if the the volume of the MP is doubled, its phase fraction must be halved. 3. One can also set constraints on the lattice when the two cells are different. However, remember that the constraints are on the reciprocal metric tensor, not on lattice parameters. Consult a crystallography book to see how they are related for the various lattices. Shubnikov Groups: a GSAS application Pnma Page 27 / 42

128 Page 28 / 42 Magnetic powder diffraction and instrumentation Paolo G. Radaelli

129 Spin Density Unit-Cell Spin Density (localised and isotropic approx.): M n atoms ~ M ( k) = mˆ µ f ( k) e u j= j j j ik r j Lattice Spin Density (simple translational symmetry): iκ r j ( = M u ( r rn ) M( k) = δ ( κ k) mˆ jµ j f j ( k) e M r) rn ~ * κ Γ n atoms Lattice Spin Density (-dimensional modulation): M r) = P rn ( r τ) M ( r r ) j ( ( ) ( l ) n u n * κ Γ l=, + j= n atoms ~ M( k) = δ κ lτ k A mˆ µ f ( k) e ilx where P( x) = Al e has a periodicity of. + l= ( r ) n atoms u ( r) = mˆ jµ jg j rj j= j= j j j iκ r Magnetic Scattering of Neutrons Neutrons are strongly scattered from magnetic moments. The scattering amplitude from an ion is of the order of γr e µ, where: γ =.9 Neutron magnetic moment in nuclear magnetons (spin + orbital). r e = cm Electron classical radius (e 2 /m e c 2 ) µ = ion magnetic moment in Bohr magnetons. For comparison, typical nuclear scattering amplitudes for neutrons are of the order of cm. Page 29 / 42

130 Magnetic Scattering of Neutrons- II Let s recall the formula for the lattice spin density and its Fourier transform: n atoms iκ r ( = Mu ( r rn ) j M( k) = δ ( κ k) mˆ jµ j f j ( k) e M r) r n ~ * κ Γ The Fourier transform M ~ (k) is called magnetic structure factor. Unlike the nuclear structure factor, it is an axial vector quantity, and it has to be combined with the other vector quantity in the problem in order to obtain the cross section, which is a scalar. The other vector quantities are the momentum transfer k (a conventional vector) and the neutron spin s n (an axial vector). j= The Magnetic Form Factor () ik r 3 q M r e dr q f ( k) = 3 q M r dr q () In the isotropic case: over a singleatom f ( k ) = f ( k) = j ( ) 2 0 k + j2( k) g From: International Tables of Crystallography, Volume C, ed. by AJC Wilson, Kluwer Ac. Pub., 998, p. 53 Page 30 / 42

131 Scattering of Neutrons from MS It is useful to introduce the quantity Q(k), known as magnetic interaction vector, and defined as: ~ Q( k) = kˆ M( k) kˆ Q(k) is the projection of the magnetic structure factor upon the plane perpendicular to the momentum transfer k. Magnetic neutron scattering cross sections only contain Q(k). In other words, scattering of neutrons through k is only determined by the components of the magnetic moments to k. Note that Q(k) can be complex. Magnetic Scattering Formulæ Polarised neutrons - polarisation analysis Non-flip Flip Total dσ d Ω ++ { n n [ ]} * * ( k) = ( γr ) sˆ Q( k) + F ( k) + sˆ Q ( k) F ( k) + Q( k) F ( k) + e [ ] * { [ ] + sˆ Q( k) Q( k) } dσ 2 * ( k ) = ( γr e ) [ sˆ Q( k) ] sˆ n n Q ( k) i n d Ω dσ d Ω { n [ ]} * * * ( k ) = ( γr ) Q( k) + F ( k) + sˆ Q ( k) F ( k) + Q( k) F ( k) + iq( k) Q( k) Unpolarised neutrons dσ d Ω e Unpol { } 2 2 ( k) = ( γ ) Q( k) 2 + F ( k) r e Page 3 / 42

132 Formulæ Explained Non-flip: Flip: In addition to the nuclear scattering, it contains the components of Q(k) parallel to the neutron spin and a magneto-structural interference term. It contains the components of Q(k) perpendicular to the neutron spin, plus an additional term which is present only if Q(k) is complex. Total: It contains the nuclear term, the module square of Q(k) and the two terms which are linear in s n. Unpolarised: It contains only the nuclear term and the module square of Q(k), since the two terms which are linear in s n cancel upon averaging. Neutron beam polarisation As we have seen, the scattering cross section depends on the initial spin direction s i. Also, in general, the final direction of the neutron spin s f is not parallel to the initial one s i. Therefore, the population of spins in a neutron beam is generally altered by magnetic scattering. One defines the neutron beam polarisation as P = s ˆ n, where ŝn is the neutron spin direction and the average is taken over all the neutrons in the beam. The transformation of the neutron polarisation upon scattering is given by: P = DP + P f i Where D is a tensor describing the effects of rotation and depolarisation and describes the creation of new polarisation. P C C Page 32 / 42

133 The simplest case-i Scattering of unpolarised neutrons from a collinear unmodulated structure. Here, κ is a reciprocal lattice vector. For collinear structures (all moments Q( κ) 2 = sin 2 ( α ) n atoms j= ( κ )e iκ r where α is the angle between κˆ and mˆ µ f j j j 2 // m) ˆ dσ d Ω Unpol { } 2 2 ( κ) = ( γ ) Q( κ) 2 + F ( κ) r e The simplest case-ii It looks like all the information is there to solve the structure even with unpolarised neutrons and powder diffraction. All the magnetic moment magnitudes are contained in Q(κ) with the appropriate phase factors and signs. Also, the information about the direction of the magnetic moments is there through the prefactor sin 2 (α). So, why bother with polarised neutrons and single-crystal techniques? Page 33 / 42

134 Magnetic Powder Diffraction Averaging of the sin 2 (α) term over the (quasi)-degenerate reflections: For Uniaxial Groups (3-fold, 4-fold, 6-fold) we can only determine the angle φ: sin 2 α = k sin ψ sin ϕ cos ψ cos ϕ ψ c φ m For Cubic Structures, the direction of the magnetic moments is undetermined: sin 2 α = 2 3 Magnetic Powder Diffractometers-I High-k range: For magnetic structure analysis, one rarely needs to go beyond sin(θ)/λ=0.5. Wavelengths > 2 Å are ideal. Low-k range: It is essential to have good coverage at low k, as many helimagnetic structures have very long periodicity. k=0.5 Å - is the minimum acceptable to do any sensible work. k=0. Å - is ideal. Resolution: it is desirable especially in structure with low crystallographic symmetry, because it enables to reduce the accidental degeneracy. Page 34 / 42

135 CW Powder Diffractometers Most magnetic structure problems are first tackled using high-intensity CW powder diffractometers (e.g., DB). The biggest advantages are the excellent coverage at low k, the high flux (that can be further enhanced through focussing) and the simplicity of the data structure. Resolution is generally quite poor. The use of high-resolution machines (e.g., D2B) is becoming more common, especially when the magnetic moments are large, the structure has low symmetry and there is an interplay between magnetism and structural properties. The High-Intensity CW Powder diffractometer DB at the ILL Page 35 / 42

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137 y=0.8 Pr 0.65 (Ca y,sr -y ) 0.35 MnO 3 y=0.7 y= Temperature (K) Bank Bank d-spacing (Å) OSIRIS Page 37 / 42